J/arithmetic.pdf.txt

Arithmetic 

Kenneth E. Iverson 

Copyright  © 2002 Jsoftware Inc. All rights reserved. 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Preface 

Arithmetic  is  the  basic  topic  of  mathematics.  According  to  the  American  Heritage 
Dictionary  [1],  it  concerns  “The  mathematics  of  integers  under  addition,  subtraction, 
multiplication, division, involution, and evolution.” 

The present text differs from other treatments of arithmetic in several respects: 

The  provision  of  simple  but  precise  definitions  of  the  counting  numbers  and  other 
notions introduced. 

The  use  of  simple  but  precise  notation  that  is  executable  on  a  computer,  allowing 
experimentation  and  providing  a  simple  and  meaningful  introduction  to  computer 
programming. 

The  introduction  and  significant  use  of  fundamental  mathematical  notions  (such  as 
vectors,  matrices,  Heaviside  operators,  and  duality)  in  simple  contexts  that  make 
them  easy  to  understand.  This  lays  a  firm  foundation  for  a  wealth  of  later  use  in 
mathematics. 

Emphasis is placed on the use of guesses by speculation and criticism in the spirit of 
Lakatos, as discussed in the treatment of proofs in Chapter 5. 

The  thrust  of  the  book  might  best  be  appreciated  by  comparing  it  with  Felix  Klein’s 
Elementary  Mathematics  from  an  Advanced  Standpoint  [2].  However,  I  shun  the 
corresponding title Arithmetic from an Advanced Standpoint because it would incorrectly 
suggest  that  the  treatment  is  intended  only  for  mature  mathematicians;  on  the  contrary, 
the  use  of  simple,  executable  notation  makes  it  accessible  to  any  serious  student 
possessing little more than a knowledge of the counting numbers. 

Like Klein, I do not digress to discuss the importance of the topics treated, but leave that 
matter to the knowledge of the mature reader and to the faith of the neophyte. 

  
 
  
Table of Contents 

Introduction ..............................................................................1 

A. Counting Numbers.......................................................................... 1 

B. Integers ........................................................................................... 2 

C. Inverses ........................................................................................... 2 

D. Domains.......................................................................................... 3 

E. Nouns and Verbs............................................................................. 3 

F. Pronouns and Proverbs.................................................................... 3 

G. Conjunctions................................................................................... 4 

H. Addition And Subtraction............................................................... 5 

I. Verb Tables ...................................................................................... 5 

J. Relations .......................................................................................... 6 

K. Lesser-Of and Greater-Of............................................................... 7 

L. List And Table Formation............................................................... 7 

M. Punctuation .................................................................................... 8 

N. Insertion.......................................................................................... 9 

O. Multiplication ................................................................................. 10 

P. Power............................................................................................... 10 

Q. Summary......................................................................................... 11 

R. On Language................................................................................... 12 

Properties of Verbs ..................................................................17 

A. Valence, Ambivalence, And Bonds................................................ 17 

B. Commutativity ................................................................................ 18 

C. Associativity ................................................................................... 18 

D. Distributivity................................................................................... 18 

E. Symmetry ........................................................................................ 19 

F. Display of Proverbs......................................................................... 20 

G. Inverses........................................................................................... 20 

H. Partitions......................................................................................... 20 

I. Identity Elements and Infinity.......................................................... 21 

J. Experimentation ............................................................................... 22 

K. Summary of Notation ..................................................................... 22 

L. On Language ................................................................................... 22 

Partitions and Selections.........................................................25 

A. Partition Adverbs............................................................................ 25 

B. Selection Verbs ............................................................................... 26 

  
 
C. Grade and Sort ................................................................................ 28 

D. Residue ........................................................................................... 28 

E. Characters........................................................................................ 29 

F. Box and Open.................................................................................. 30 

G. Summary of Notation ..................................................................... 31 

H. On Language .................................................................................. 31 

Representation of Integers ......................................................33 

A. Introduction .................................................................................... 33 

B. Addition .......................................................................................... 34 

C. Multiplication.................................................................................. 35 

D. Normalization ................................................................................. 37 

E. Mixed Bases.................................................................................... 39 

F. Experimentation .............................................................................. 40 

G. Summary of Notation ..................................................................... 41 

Proofs ........................................................................................43 

A. Introduction .................................................................................... 43 

B. Formal and Informal Proofs............................................................ 47 

C. Proofs and Refutations.................................................................... 48 

D. Proofs.............................................................................................. 50 

Logic..........................................................................................57 

A. Domain and Range ......................................................................... 57 

B. Propositions .................................................................................... 58 

C. Booleans ......................................................................................... 58 

D. Primitives........................................................................................ 60 

E. Boolean Dyads ................................................................................ 61 

F. Boolean Monads.............................................................................. 62 

G. Generators....................................................................................... 62 

H. Boolean Primitives.......................................................................... 63 

I. Summary of Notation ....................................................................... 63 

Permutations ............................................................................65 

A. Introduction .................................................................................... 65 

B. Arrangements.................................................................................. 67 

D. Products of Permutations................................................................ 69 

E. Cycles.............................................................................................. 70 

F. Reduced Representation .................................................................. 71 

G. Summary of Notation ..................................................................... 72 

Classification and Sets ............................................................75 

  
A. Introduction .................................................................................... 75 

B. Sets.................................................................................................. 78 

C. Nub Classification........................................................................... 80 

D. Interval Classification..................................................................... 80 

E. Membership Classification.............................................................. 81 

F. Summary of Notation ...................................................................... 83 

Polynomials ..............................................................................85 

A. Introduction .................................................................................... 85 

B. Sums and Products.......................................................................... 86 

C. Roots ............................................................................................... 87 

D. Expansion ....................................................................................... 88 

E. Graphs And Plots ............................................................................ 89 

F. Real And Complex Numbers .......................................................... 89 

G. General Expansion.......................................................................... 92 

H. Slopes And Derivatives .................................................................. 93 

I. Derivatives of Polynomials .............................................................. 96 

J. The Exponential Family................................................................... 96 

K. Summary Of Notation..................................................................... 99 

L. On Language ................................................................................... 99 

References ................................................................................107 

  
1 

Chapter 
1 

Introduction 

A. Counting Numbers 

The list 1 2 3 4 5 6 7 8 9 10 11 12 shows the first dozen counting numbers, and 
any reader of this book could extend the list to tedious lengths. Although this definition 
by example captures the basic idea, it fails to address related questions such as: 

1.  Do counting numbers continue forever? 

2.  Are there other numbers that precede the first counting number? 

3.  Are there other numbers between the counting numbers or elsewhere? 

These questions were addressed a century ago by Peano, who began by introducing the 
notion of a successor “operation” which, when applied to any counting number, produced 
its successor. For example, successor 3 would produce 4.  

We will denote the successor operation by the two-character word >:  . For example: 

  >: 3 
4 

  >: 999 
1000 

The  foregoing  is  an  example  of  dialogue  with  the  computer.  Because  the  notation used 
here  (and  throughout  the  book)  can  be  executed  by  a  computer  provided  with  the 
language J (available from website jsoftware.com), every expression used can be tested 
by executing it, as can related expressions that the reader may wish to experiment with. 
For example, one might apply the successor to lists of counting numbers as follows: 

   >: 1 2 3 4 5 6 7 8 9 10 11 12 
2 3 4 5 6 7 8 9 10 11 12 13 

   >: 2 4 6 8 10 
3 5 7 9 11 

  
 
 
 
 
 
 
2  Arithmetic 

Is there a  last or largest counting number? Peano answered this by asserting that every 
counting number has a distinct successor, thus introducing the idea of an unbounded or 
infinite list of counting numbers. 

B. Integers 

Since  7  is  the  successor  of  6,  we  may  also  say  that  6  is  the  predecessor  of  7,  and 
introduce a predecessor operation denoted by <:  . For example: 

   <:3 5 7 9 11 
2 4 6 8 10 

   >:2 4 6 8 10 
3 5 7 9 11 

It  would  be  convenient  if  the  predecessor  (like  the  successor)  applied  to  all  counting 
numbers,  but  since  1  is  the first counting number, its predecessor cannot be a counting 
number. We therefore introduce a wider class of numbers, in which every member has a 
predecessor as well as a successor. Thus: 

   <: 1 
0 
   <: 0 
_1 
   <: _1 
_2 

This  wider  class  of  numbers  is  called  the  integers,  and  includes  zero  (0),  as  well  as 
negative numbers (_1 _2 _3 etc.). 

It is helpful to form the habit of looking up any new technical term in a good dictionary; 
even  if  the  term  is  already  familiar,  its  etymology  often  provides  useful  insight.  For 
example, in the American Heritage Dictionary (a dictionary to be recommended because 
of its method of treating etymology) the definition of integer refers to the Indo-European 
root tag that means “to touch; handle”. This with the prefix in- (meaning not) implies that 
an integer is untouched, or whole; in contrast to one that is “fractured”, like one of the 
fractions one-half, one-quarter, etc. 

Similarly, the word infinite introduced in Section A will be found to mean not (in) finite, 
or without finish. 

C. Inverses 

The predecessor operation (<:) is said to be the inverse of the successor (>:) because it 
“undoes”  its  work. For example,  <:>:  8 yields  8, and the same relation holds for any 
integer. Thus: 

   >:1 2 3 4 5 6 
2 3 4 5 6 7 

<:>:1 2 3 4 5 6 

1 2 3 4 5 6 

In the original definition the successor applied only to the counting numbers. We now re-
define it to apply to all integers by defining it as the inverse of predecessor. For example: 

  
 
 
 
 
 
 
Chapter 1  Introduction   3 

   >:<: _3 _2 _1 0 1 2 
_3 _2 _1 0 1 2 

D. Domains 

The successor >: defined in Section A applied only to counting numbers, and they would 
be said to be its domain (over which it “ruled”). In defining the predecessor in Section B 
it became necessary to extend its domain to the integers, that also included zero and the 
negative numbers. By re-defining the successor as the inverse of the predecessor, we also 
extended its domain to the integers. 

We  will  find  that  the  introduction  of  further  operations  (such  as  the  inverse  of 
“doubling”)  will  require  further  extensions  of  domains.  However,  to  keep  the 
development simple, we will restrict attention to simple domains as far as possible. 

E. Nouns and Verbs 

The  successor  operation  >:  can  be  said  to  “act  upon”  a  counting  number  to  produce  a 
result, and is therefore analogous to an “action word” or verb in English. Similarly, the 
numbers to which the verb >: applies are analogous to nouns in English. 

We will soon see that the terms verb and noun lead to further important analogies with 
adverbs,  conjunctions,  and  other  parts  of  speech  in  English.  We  will  therefore  adopt 
them,  even  though  other  terms  (function,  operator,  and  variable)  are  more  commonly 
used in mathematics. However, function will sometimes be used as a  synonym for verb. 

F. Pronouns and Proverbs 

Consider the following use of the pronoun it : 

   it=: 1 2 3 4 5 6 
   <: it 
0 1 2 3 4 5 

   >:<: it 
1 2 3 4 5 6 

The  copula  =:  behaves  like  the  copulas  is  and  are  in  English,  and  the  first  sentence 
would be read aloud as “it is the list of counting numbers 1 2 3 4 5 6” or as “it is 1 
2 3 4 5 6”. 

In English the names used for pronouns are restricted to a very few, such as it, he, and 
she; they are not so restricted here. For example: 

   zero=: 0    
   neg=: _1 _2 _3 
   list6=: it 
   list6,zero,neg 
1 2 3 4 5 6 0 _1 _2 _3 

  
 
 
 
 
 
 
4  Arithmetic 

A proverb is used to stand for a verb, just as a pronoun is used to stand for a noun. (The 
word proverb in this sense is found only in larger dictionaries.) For example: 

   increment=: >: decrement=: <: 
   increment list6,zero,neg 
2 3 4 5 6 7 1 0 _1 _2 

   inc=: increment 
   inc list6 
2 3 4 5 6 7 

G. Conjunctions 

The  phrase  Run  and  hide  expresses  an  action  performed  as  a  sequence  of  two  simpler 
actions,  and  in  it  the  word  and  is  said  to  be  a  copulative  conjunction.  We  will  use  the 
symbol @ to denote an analogous conjunction. For example: 

   add3=: >: @ >: @ >: 
   add3 1 2 3 4 5 6 
4 5 6 7 8 9 

   identity=: <: @ >: 
   identity 1 2 3 4 5 6 
1 2 3 4 5 6 

Although  the  verb  identity  defined  above  makes  no  change  to  its  argument,  it  is  an 
important verb, so important that it is given its own symbol. Thus: 

   ] 1 2 3 4 5 6 
1 2 3 4 5 6 

Although  a  verb  for  the  twelfth  successor  could  be  expressed  by  repeated  use  of  @,  it 
would be tedious, and we introduce a second conjunction illustrated below: 

   list=: 1 2 3 4 5 6 
   >:^:3 list 
4 5 6 7 8 9 

   >:^:12 list 
13 14 15 16 17 18 

   <:^:6 list 
_5 _4 _3 _2 _1 0 

The conjunction ^: is called the power conjunction; it applies its left argument (the verb 
to its left) the number of times specified by its noun right argument. 

  
 
 
 
 
 
 
 
 
 
 
 
H. Addition And Subtraction 

The examples of the preceding section illustrate the fact that if n is any counting number, 
then the verb >:^:n adds n  to its argument, and <:^:n subtracts n. 

Chapter 1  Introduction   5 

For example : 

   n=: 5 
   abc=: 10 11 12 13 14 15 
   >:^:n abc 
15 16 17 18 19 20 

   <:^:n abc 
5 6 7 8 9 10 

   abc+n 
15 16 17 18 19 20 

abc-n 
5 6 7 8 9 10 

The  last  two  examples  introduce  the  notation  commonly  used  for  addition  and 
subtraction,  and  the  whole  set  of  examples  essentially  defines  them  in  terms  of  the 
simpler successor and predecessor of Peano. 

I. Verb Tables 

Two lists can be added and subtracted as illustrated below: 

   a=: 0 1 2 3 4 5 
   b=: 2 3 5 7 11 13 
   a+b 
2 4 7 10 15 18 

a-b 

_2 _2 _3 _4 _7 _8 

   a+a 
0 2 4 6 8 10 
   a-a 
0 0 0 0 0 0 

   a +/ b 
2 3  5  7 11 13 
3 4  6  8 12 14 
4 5  7  9 13 15 
5 6  8 10 14 16 
6 7  9 11 15 17 
7 8 10 12 16 18 

   a +/ a 
0 1 2 3 4  5 
1 2 3 4 5  6 
2 3 4 5 6  7 
3 4 5 6 7  8 
4 5 6 7 8  9 
5 6 7 8 9 10 

  
 
 
 
 
 
 
 
6  Arithmetic 

The last two examples show addition tables that add each item of the first argument to 
each item of the second in a systematic manner. The verb +/ is formed by applying the 
adverb / to the verb + , and is usually referred to as the verb “plus table”. The adverb / 
applies  uniformly  to  other  verbs,  and  we  can  therefore  produce  subtraction  tables  as 
follows: 

  a-/a 
0 _1 _2 _3 _4 _5 
1  0 _1 _2 _3 _4 
2  1  0 _1 _2 _3 
3  2  1  0 _1 _2 
4  3  2  1  0 _1 
5  4  3  2  1  0 

b-/1 2 

 1  0 
 2  1 
 4  3 
 6  5 
10  9 
12 11 

To make clear the meaning of a verb table, draw a vertical line to its left and write the left 
argument vertically to the left of it; draw a horizontal line above the table, and enter the 
right argument horizontally above it. We can produce such an annotated display of a verb 
table by using the adverb table instead of /, as follows: 

   a +table b 
+-+---------------+ 
| |2 3  5  7 11 13| 
+-+---------------+ 
|0|2 3  5  7 11 13| 
|1|3 4  6  8 12 14| 
|2|4 5  7  9 13 15| 
|3|5 6  8 10 14 16| 
|4|6 7  9 11 15 17| 
|5|7 8 10 12 16 18| 
+-+---------------+ 

   a-table a 
+-+----------------+ 
| |0  1  2  3  4  5| 
+-+----------------+ 
|0|0 _1 _2 _3 _4 _5| 
|1|1  0 _1 _2 _3 _4| 
|2|2  1  0 _1 _2 _3| 
|3|3  2  1  0 _1 _2| 
|4|4  3  2  1  0 _1| 
|5|5  4  3  2  1  0| 
+-+----------------+  

J. Relations 

Any two integers a and b are related in certain simple ways: a precedes (or is less than) 
b; a equals b; or a follows (or is greater than) b. We introduce the verbs < and = and > 
whose  results  show  whether  the  particular  relation  holds  between  the  arguments.  For 
example: 

   1<3 
1 

1=3 

0  

1>3 

0 

   a=: 1 2 3 4 5 
   b=: 6-a 
   b 

  
 
 
 
 
 
 
Chapter 1  Introduction   7 

5 4 3 2 1 

   a<b 
1 1 0 0 0 

   a=b 
0 0 1 0 0 

   a</b 
1 1 1 1 0 
1 1 1 0 0 
1 1 0 0 0 
1 0 0 0 0 
0 0 0 0 0 

   a=/b 
0 0 0 0 1 
0 0 0 1 0 
0 0 1 0 0 
0 1 0 0 0 
1 0 0 0 0 

a>b 
0 0 0 1 1 

a>/b 
0 0 0 0 0 
0 0 0 0 1 
0 0 0 1 1 
0 0 1 1 1 
0 1 1 1 1 

A  result  of  1  indicates  that  the  relation  holds,  and  0  indicates  that  it  does  not;  it  is 
reasonable to read the ones and zeros aloud as “true” and “false”. The final example is a 
greater-than table. 

K. Lesser-Of and Greater-Of  

The lesser of (or minimum of) two arguments is the one that precedes (or perhaps equals) 
the other; the verb <. yields the lesser of its arguments. For example: 

b 

5 4 3 2 1 

a>.b 
5 4 3 4 5 

   a 
1 2 3 4 5 

   a<.b 
1 2 3 2 1 

   a<./b 
1 1 1 1 1 
2 2 2 2 1 
3 3 3 2 1 
4 4 3 2 1 
5 4 3 2 1 

L. List And Table Formation 

Although  any  list  can  be  specified  by  listing  its  members,  certain  lists  can  be  specified 
more conveniently. The  integers verb  i. produces lists or tables of integers (beginning 
with zero) that are convenient in producing verb tables. For example : 

  ] a=:i. 5 
0 1 2 3 4 

   a<./a 

  
 
 
 
 
 
 
 
 
 
 
8  Arithmetic 

0 0 0 0 0 
0 1 1 1 1 
0 1 2 2 2 
0 1 2 3 3 
0 1 2 3 4 

   4-a 
4 3 2 1 0 

   1+a 
1 2 3 4 5 
   i. _5 
4 3 2 1 0 
   i.3 4 
0 1  2  3 
4 5  6  7 
8 9 10 11 

The verb # replicates its right argument the number of times specified by the left: 

   3#5 
5 5 5 

   5#3 
3 3 3 3 3 

   2 3 4 # 6 7 8 
6 6 7 7 7 8 8 8 8 

   b=: _2 + i. 5 
   b 
_2 _1 0 1 2 

   c=:b>0 
   c 
0 0 0 1 1 
   c#b 
1 2 

The verb $ “shapes” its right argument, using cyclic repetition of its items as needed: 

   8$2 3 5 
2 3 5 2 3 5 2 3 

3 4$2 3 5 

2 3 5 2 
3 5 2 3 
5 2 3 5 

M. Punctuation 

Although the two sentences: 

The teacher said he was stupid 

The teacher, said he, was stupid 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
differ only in punctuation, they differ greatly in meaning. 

Arithmetic sentences may also be punctuated (by paired parentheses) as illustrated below: 

Chapter 1  Introduction   9 

   (8-3)+4 
9 
   8-(3+4) 
1 
   8-3+4 
1 

The  last  sentence  illustrates  the  behaviour  in  the  absence  of  parentheses:  in  effect,  the 
sentence is evaluated from right to left or, equivalently, the right argument of each verb is 
the value of the entire phrase to its right. 

Punctuation makes possible many useful expressions. For example: 

   c=: 2 7 1 8 2 8 
   (c=2)#c 
2 2 

   ((c=2)>.(c=8))#c 
2 8 2 8 

   (c<2)>.(c=2) 
1 0 1 0 1 0 

The last sentence can be read as “c is less than or equal to 2”. It is equivalent to the verb 
<: in the expression c<:2. 

The  beginner  is  advised  to  use  fully-parenthesized  sentences  even  though  some  of  the 
parentheses  are  redundant.  Thus,  write  (c<2)>.(c=2)  even  though  (c<2)>.c=2  is 
equivalent. 

N. Insertion 

   a=: 2 7 1 8 2  
   2+7+1+8+2 
20 
   +/a 
20 

The foregoing sentences illustrate the fact that the adverb / produces a verb that “inserts” 
its verb left argument between the items of the argument of the resulting verb +/ . Insert 
applies equally to other verbs. For example:  

2>.7>.1>.8>.2 

8 

   >./a 
8 

   sum=:+/ 

   max=:>./ 

  
 
 
 
 
 
 
 
 
 
 
10  Arithmetic 

   min=:<./ 

   sum a 
20 

   spread=: (max a)-(min a) 
    range=: (min a)+i. >:spread 
   range 
1 2 3 4 5 6 7 8 

O. Multiplication 

   m=:3 
   n=:5 
   n#m 
3 3 3 3 3 

   +/n#m 
15 

The  final  result  above  is  clearly  the  product  of  m  and  n,  and  the  sentences  essentially 
define  multiplication  in  terms  of  repeated  addition.  In  mathematics  the  product  verb  is 
denoted in a variety of ways; we will use * as in: 

   m*n 
15 

   dig=: 1+i. 6 
   dig 
1 2 3 4 5 6 

odds=: 1+2*i. k=: 6 
odds 

1 3 5 7 9 11 

   */dig 
720 
   !#dig 
720 

+/odds 

36 

 k*k 

36 

The  last  two  sentences  on  the  left  illustrate  the  definition  of  a  new  verb,    factorial, 
denoted by ! . 

P. Power 
   m=: 3 
   n#m  
3 3 3 3 3 

n=: 5 
   */n#m 
243 

The final result above is called the nth power of m, or m to the power n. Comparison with 
Section O will show that power is defined in terms of multiplication in the same way that 
multiplication is defined in terms of addition. 

  
 
 
 
 
 
 
 
 
 
 
 
In most math texts there is no symbol for power, it being denoted by showing the second 
argument  as  a  superscript.  We  will  adopt  the  symbol  ^  used  by  de  Morgan  [3]  about  a 
century ago. For example: 

Chapter 1  Introduction   11 

   m^n  
243 

3^5 

243 

   (3^5)*(3^2) 
2187 

3^(5+2) 

2187 

As suggested by the equivalence of the last two sentences, (a^b)*(a^c) is equivalent to 
a^(b+c). The reason for this can be seen by substituting the definition of power given 
above:       

   (3^5)*(3^2) 
2187 

(*/5#3)*(*/2#3) 

2187 

   (5+2)#3 
3 3 3 3 3 3 3 

*/(5+2)#3 

2187 

Q. Summary 

The main results of this chapter may be summarized as follows: 

1.  The  idea  of  the  counting  numbers  is  formalized  and  extended  to  infinity  by 
introducing the notion that every counting number has a successor; it is extended 
to include zero and negative numbers by introducing the notion of predecessor, 
inverse to successor. 

2.  Symbols  are  introduced  to  denote  successor  and  predecessor  (>:  and  <:); 
because they specify actions they are called verbs, and the integers they act upon 
are called nouns. 

3.  The copula =: is introduced to assign a name (called a pronoun) to a noun or list 

of nouns and to assign a name (called a proverb) to a verb. 

4.  Conjunctions  (@  and  ^:)  are  introduced  to  define  verbs  that  are  specified  by  a 

sequence of simpler verbs. 

5.  Addition is defined in terms of a sequence of successors; subtraction is defined in 

terms of predecessors. 

6.  Verb tables are introduced to display the behaviour of addition, subtraction, and 
other verbs that apply to two arguments, such as relations (< = >) and minimum 
and maximum (<. >.). 

7.  Parentheses are introduced as punctuation, that is, to specify the order in which 

phrases in a sentence are to be interpreted. 

8        An  adverb  called  insert  (denoted  by  /)  is  introduced  to  insert  a  verb  between 
items  of  a  list  argument,  and  +/  is  used  with  replication  (#)  to  define 
multiplication  in  terms  of  repeated  addition;  power  is  defined  in  terms  of 
repeated multiplication. 

We  will  now  summarize  all  of  the  notation  used.  This  summary  may  be  useful  for 
reference, but because related symbols are used for related ideas, it should also be studied 

  
 
 
 
 
12  Arithmetic 

for  mnemonic  aids.  Succeeding  chapters  conclude  with  similar  summaries  of  notation, 
and all notation is available from the J Dictionary discussed in Book 1. 

The  table  shows  the  verbs  in  three  columns,  each  headed  by  the  final  character  (dot  or 
colon) of the verbs in that column: the first row shows Less than (<) in the first column, 
Lesser of (<.) in the second, and Predecessor (<:) in the third: 

Verbs And Copula 

. 

: 

<  Less than 

Lesser of (Min) 

Predecessor 

>  Greater than 

Greater of (Max) 

Successor 

Copula 

=  Equals 

+  Add 

- 

Subtract 

*  Multiply 

^ 

! 

Power 

Factorial 

]  

Identity 

Replicate 

Shape 

Catenate 

# 

$ 

, 

i 

Integers 

Adverbs 

/  Insert (when used with one noun argument, as in +/b)  

  Table (when used with two noun arguments, as in a+/b) 

Conjunctions 

@  Atop (defines a verb by a sequence, as in >:@>:@>:) 

^: Power (>:^:3 is >:@>:@>:) 

In conventional math, the symbol - denotes subtraction when used with two arguments 
(a-b) and negation when used with one (-b). We will adopt this usage, defining -b by 
0-b. 

The thoughtful reader may have noticed such usage in this chapter: the verbs produced by 
the  adverb  /  (as  shown  above),  and  the  <:  used  for  predecessor  throughout,  but  used 
dyadically (that is, with two arguments) for Less or equal in Section M. This ambivalent 
use of verbs is discussed fully in Chapter 2. 

R. On Language 

Notation, the term normally used to refer to the mode of expression in math, is defined 
(in  the  AHD)  as  “A  system  of  figures  or  symbols  used  in  specialized  fields  ...  ”.  An 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 1  Introduction   13 

executable notation such as that used here is normally called a programming language; 
we will use the terms notation and language interchangeably.  

Programming languages are commonly taught in specific courses, prerequisite to courses 
in  topics  that  employ  them.  In  mathematics,  on  the  contrary,  notation  is  not  taught  as 
such,  but  is  introduced  in  passing  as  required  by  the  subject.  The  same  approach  is 
adopted in this text. 

Any reader interested in using the notation in topics other than those treated here should 
consult Section 9 L. 

In  a  math  course  there  is  little  reason  for  a  student  to  be  curious  or  concerned  about 
notation  that  has  not  yet  been  used.  In  using  a  programming  language  the  situation  is 
somewhat  different;  a  student  who  already  knows  something  of  the  possibilities  of 
computer  programming  may  feel  frustrated  at  not  knowing  what  symbols  to  use  for 
operations that she knows must be available in the language. 

There  are  several  avenues  open  to  the  student  who  may  be  more  interested  in  the 
language than in the treatment of arithmetic: 

1.  Press key F1 in the top row to display the vocabulary of J. Then click the mouse 
on  any  desired  entry  in  the  vocabulary  to  display  its  definition.  Press  Esc  to 
remove the display. 

2.  Use the computer to experiment with various facilities, and therefore to explore 

their definitions. 

3.  Range ahead to the On Language sections that conclude Chapters 2 and 9. 

Exercises 

In exercises first write (or at least sketch out) the result of each sentence without using 
the computer; then enter the sentence on the computer to check your answer. 

In  using  the  computer,  it  will  be  more  efficient  if  you  familiarize  yourself  with  the 
available editing facilities. In particular, these allow you to revise entries being prepared, 
and to recall earlier entries for re-entry. Also learn to use expressions such as: 

   names 0 

To display the names used for pronouns 

   names 1 

To display the names used for adverbs 

   names 2 

To display the names used for conjunctions 

   names 3 

To display the names used for proverbs 

   erase <'abc' 

To erase the name abc 

Letters such as A and B in the labels below indicate the sections to which the associated 
experiments are relevant. Refer back to these sections for any needed help: 

A1  >:12345 

>:1 2 3 4 5 

>:>:>:>:1 2 3 4 5 

  
 
 
 
 
 
 
 
 
 
14  Arithmetic 

B1  <: _12345 

<:_1 _2 _3 _4 _5 

<:<:<:<:1 2 3 4 5 

<:<:>:>:1 2 3 4 5 

>:<:>:<:1 2 3 4 5 

F1  a=:1 2 3 

b=:4 5 

>:a 

a,b 

>:a,b 

F2  z=:0 

n=:_5 _4 _3 _2 _1 

n,z,a,b 

b,a,z,n 

F3 

wax=: >: 

wane=:<: 

wax wax wane n,z,a,b 

G1  list=:1 2 3 4 5 

right=:>:@>: 

left=:<:@<: 

right list 

left list  

left right list 

] list 

G2  decade=:>:^:10 

decade list 

century=:decade^:10 

century list 

>:^:10^:10 list 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
G3  First  review  the  discussion  of  inverses  in  Section  C.  Then  enter  the  following 
sentences  on  the  computer,  observe  their  results,  and  try  to  state  the  effect  of the 
power conjunction with negative right arguments: 

Chapter 1  Introduction   15 

>:^:_1 list 

<:^:_1 list 

>:^:_3 list 

decade^:_1 list 

decade^:2 decade^:_2 list 

 I1 

Reproduce on the computer the last two tables of Section I. 

J1 

The  verbs  over  and  by  used  in  the  following  sentences  were  defined  and 
illustrated  in  Section  I.  As  usual,  first  sketch  the  result  of  each  sentence  by  hand 
before entering it on the computer: 

d=: 0 1 2 3 4 

d by d over d</d 

d by d over d=/d 

d by d over d+/d 

d by d over d-/d 

J2  Repeat Exercise J1 using the list e=:_3 _2 _1 0 1 2 3 instead of the list d. 

K1  Repeat Exercises J1 and J2 for the verbs >. and <., that is, for tables of maximum 

and minimum. 

M1  An integer such as 14 that can be written as the sum of some integer with itself is 
called  an  even  number;  a  number  such  as  7  that  cannot  is  called  odd.  Write  an 
expression using the verb i. to produce the first twenty even numbers. Do not look 
at the answer below until you have tested your answer on the  computer. 

Answer:   (i.20)+(i.20)  

M2  Write an expression for the first 20 odds.  

N1  Review  Section  M  and  note  that  the  unparenthesized  sentence  2-7-1-8-2  is 
equivalent to 2-(7-(1-(8-2))) . Then evaluate the sentence and verify that your 
result agrees with -/2 7 1 8 2. 

Evaluate and compare the results of the following sentences: 

-/2 7 1 8 2  

(+/2 1 2)-(+/7 8) 

Then state in simple terms what the verb -/ produces, and test your statement on 
other lists (including lists with both odd and even numbers of items). 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
16  Arithmetic 

Answer:  -/  list produces the  alternating sum, the sum of every other item of 
the list diminished by the sum of the remaining items.  

O1  Construct the multiplication table produced by the sentence  (2+i.9)*/(2+i.9) 
and  observe  that  its  largest  item  is  100.  Note  that  the  table  cannot  contain  prime 
numbers (which cannot be products of positive integers other than themselves and 
1). Examine the table to determine all of the primes up to 9. 

P1  b=:i.7 

b by b over b^/b 

a=:b-3 

     a by b over a^/b 

  
  
 
 
 
 
17 

Chapter 
2 

 Properties of Verbs 

A. Valence, Ambivalence, And Bonds 

In the phrases a-b and a<:b and a+/b the verbs “bond to” two arguments and (adopting 
an analogous term from chemistry) we say that in this context the verbs have valence 2; 
in the expressions -b and <:b and +/b the same verbs have valence 1. 

From  these  examples  it  is  clear  that  the  verbs  are  ambivalent,  the  valence  being 
determined  by  the  context  in  which  they  are  used.  We  also  say  that  a  verb  used  with 
valence 1 is used  monadically, or is a  monad; a verb used with valence 2 is a dyad. 

In the phrase 3&* the conjunction & bonds the noun 3 to the verb * to produce a monad. 
Thus: 

   triple=: 3&* 
   triple a=: 1 2 3 4 
3 6 9 12 
   square=: ^&2 
   square a 
1 4 9 16 

   ^&3 a 
1 8 27 64 

Although a is the list 1 2 3 4, it should be noted that the phrase ^&3 1 2 3 4 is not 
equivalent to ^&3 a, because the sequence 3 1 2 3 4 is treated as a single list that is 
bonded to ^ to form a verb. However, ^&3 (1 2 3 4) and ^&3 a are equivalent. 

The bond conjunction is extremely prolific because its use with any dyad d generates two 
families of monads, one using left bonding (n&d) and one using right bonding (d&n). For 
example,  with  right  bonding  the  verb  ^  produces  the  square,  cube,  and  higher  powers; 
with left bonding it produces exponential verbs. 

The conjunction @ introduced in Section 1 G composes two verbs, as in i.@- 3 to yield 
2 1 0; the verb i.@- also has a dyadic meaning, as in 8 i.@- 3 to yield 0 1 2 3 4. 
In general, v1@v2 b is equivalent to v1 v2 b, and a v1@v2 b is equivalent to v1 (a 
v2 b). In effect, the monad v1 is applied “atop” the dyad v2, and the conjunction @ 
(denoted by the commercial at symbol) is called atop.  

  
 
18  Arithmetic 

B. Commutativity 

The  dyads  +  and  *  yield  the  same  results  if  their  arguments  are  interchanged  or 
“commuted”, and they are therefore said to be commutative. For example: 

   3+5 
8   

5+3 

8 

(3*5)=(5*3) 

1 

The dyad produced by the commute or cross  adverb ~ “crosses” the bonds of the verb to 
which it is applied. Moreover, the monad produced by ~ duplicates its single argument. 
For example:  

5-3 

2 

^~3 

27 

   3-~5 
2 

  +~3 
6 

   */~i.5 
0 0 0  0  0 
0 1 2  3  4 
0 2 4  6  8 
0 3 6  9 12 
0 4 8 12 16 

C. Associativity 

Compare  the  results  of  the  following  pairs  of  sentences,  which  differ  only  in  the 
“associations” produced by different punctuations: 

   (4+3)+(2+1) 
10 
   (4-3)-(2-1) 
0 
  (4>.3)>.(2>.1) 
4 

4+((3+2)+1) 

10 

4-((3-2)-1) 

4 

4 

4>.((3>.2)>.1) 

  (4*3)*(2*1) 
24  

4*((3*2)*1) 

24 

  (4^3)^(2^1) 
4096 

4^((3^2)^1) 

262144 

Those verbs (+ >. and *) that yield the same results are examples of associative verbs; 
the others are non-associative. 

D. Distributivity 

The monad >: is said to distribute over the dyad <. because a sentence such as (>:7) 
<.  (>:4)  has  the  same  result  as  the  corresponding  sentence  >:(7<.4)  in  which  the 

  
 
 
 
 
 
 
 
 
 
monad  >:  is  “distributed  over”  the  result  of  the  dyad  <.  .  Observe  the  further  tests  of 
distributivity: 

Chapter 2 Properties of Verbs  19 

   a=:7 
   b=:4 
   triple=: *&3 
   (triple a) + (triple b) 
33 

   (triple a) - (triple b) 
9 

   (*&3 a) <. (*&3 b) 
12 

   (-&3 a) <. (-&3 b) 
1    

   (3&- a) <. (3&- b) 
_4 

triple (a+b) 

triple (a-b)  

*&3 (a<.b)  

-&3 (a<.b)  

3&- (a<.b)  

33 

9 

12 

1 

_1 

In the last two pairs of sentences it appears that although the monad -&3 (which subtracts 
3  from  its  argument)  distributes  over  minimum,  the  monad  3&-  (which  subtracts  its 
argument from 3) does not. 

This  point  is  made  to  show the pitfall in a common practice in math, where it is stated 
that  the  dyad  *  distributes  over  addition,  rather  than  stating  (as  we  do  here)  that  the 
family *&n of right bonds of * distributes over addition. 

Because  *  is  commutative,  the  left  bond  c&*  is  equivalent  to  the  right  bond  *&c,  and 
both  distribute  over  addition.  However,  in  the  case  of  a  non-commutative  verb  such  as 
subtraction,  it  is  possible  that  a  right  bond  with  a  given  dyad  distributes  while  the 
corresponding left bond does not. In such a case it is clearly incorrect to say that the dyad 
distributes,  and  one  is  led  to  statements  such  as  “-  distributes  to  the  right  over 
minimum”. 

A linear verb (to be discussed further in Chapter 9) is one that distributes over addition. 

E. Symmetry 

If a dyad d (such as + or * or >.) is both associative and commutative, then the monad 
d/  produced  by  insertion  is  said  to  be  symmetric,  because  it  produces  the  same  result 
when the argument list to which it applies is re-ordered or permuted. For example: 

   a=: 1 2 3 4 5 
   b=: 3 1 5 2 4 
   +/a 
15  

   */a 
120 

+/b 

15 

*/b 

120 

   >./a 

>./b 

 
  
 
 
 
 
 
 
 
 
20  Arithmetic 

3 

  -/a 
3 

3 

9 

-/b 

F. Display of Proverbs 

If a proverb is entered alone (that is, without arguments), its representation is displayed. 
For example, if the proverbs of Sections F and G of Chapter 1 are already defined, then: 

   increment 
>: 

   add3 
>:@>:@>: 

   identity 
<:@>: 

G. Inverses 

Review  the  discussion  of  inverses  in  Section  C  and  Exercise  G3  of  Chapter  1.  Then 
observe the results of the following uses of inversion: 

   a=:0 1 2 3 4 5 
   >:^:_1 a 
_1 0 1 2 3 4 

   >:^:_1 
< 
   +&3^:_1 a 
_3 _2 _1 0 1 2 

   +&3^:_1 
-&3 

   -&3^:_1 a 
3 4 5 6 7 8 

   3&-^:_1 a 
3 2 1 0 _1 _2 

   3&- 3&-^:3 a 
0 1 2 3 4 5 

   3&-^:_1 
3&- 

H. Partitions 

The sum of a list (+/list) is equal to the sum of sums over parts of the list, and a similar 
relation holds for some other verbs such as */ and >./ . For example: 

  
 
 
 
 
 
 
 
 
 
 
 
Chapter 2 Properties of Verbs  21 

   +/3 1 4 1 5 9 
23 

(+/3 1)+(+/4 1 5 9) 

23 

   */3 1 4 1 5 9 
540 

(*/3 1)*(*/4 1 5 9) 

540 

   >./3 1 4 1 5 9 
9 

9 

(>./3 1)>.(>./4 1 5 9) 

These relations can be expressed more clearly in terms of the truncation verbs take ({.) 
and drop (}.). Thus: 

   a=:3 1 4 1 5 9 
   2{.a 
3 1 

   2}.a 
4 1 5 9 

   (+/2{.a)+(+/2}.a) 
23 

+/a 

23 

   (*/2{.a)*(*/2}.a) 
540 

*/a 

540 

   (+/6{.a)+(+/6}.a) 
23 

   (*/6{.a)*(*/6}.a) 
540 

The last two examples are interesting because the list 6}.a is empty, yet the results of +/ 
and */ upon it are such as to maintain the identities seen for the other cases. Thus: 

  +/6}.a  */6}.a 
0  1 

This matter is explored further in the succeeding section. 

I. Identity Elements and Infinity 

It is easy to verify that the monads 0&+ and 1&* and -&0 are identity verbs that produce 
no change in their arguments. A noun that bonds with a dyad to form an identity verb is 
said to be an identity element of that dyad. Thus, 1 is the identity element of *, and 0 is 
the identity element of + and of - . 

Although -&0 is an identity, 0&- is not. We may therefore say more precisely that 0 is a 
right identity of - . The same is true for other non-commutative verbs. Thus, 1 is a right 
identity of ^ (power). 

 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
22  Arithmetic 

To  ensure  that  identities  of  the  form  (+/a)=(+/k{.a)+(+/k}.a)  remain  true  when 
one of the lists is empty, we define the result of d/b to be the identity element of d if the 
list b is empty. 

Does the dyad <. (minimum) possess an identity element? If h were a huge number (such 
as  10^9)  then  it  would  serve  for  all  practical  purposes  as  the  identity  element  of 
minimum. However, since there is no largest number among the integers, we must again 
extend the domain by adding a new element, denoted by _ and called infinity. To provide 
an identity for maximum we also add a negative infinity denoted by __ . We will refer to 
the resulting domain as integers+. Thus: 

   <./0#0 
_ 

>./i.0 

__ 

J. Experimentation 

In  experimenting  with  expressions  on  the  computer  you  will  find  that  many  verbs, 
adverbs, and conjunctions have meanings that are more general than the definitions given 
in the text.  For example: 

   halve=: 2&*^:_1 
   halve 2 4 6 8 10 
1 2 3 4 5 

   sqr=:*~ 
   sqrt=: sqr^:_1 
   sqrt 1 4 9 16 25 
1 2 3 4 5 

 halve 1 2 3 4 5 

0.5 1 1.5 2 2.5 

sqrt 1 2 3 4 5   

1 1.41421 1.73205 2 2.23607 

   sqrt - 1 2 3 4 5 
0j1 0j1.41421 0j1.73205 0j2 0j2.23607 

Some  of  the  results  of  these  experiments  are  fractions  and  complex  numbers  that  lie 
outside the domain of integers treated thus far. There is no harm in experimenting further 
with any that interest you, but do not spend too much time on baffling matters that will be 
treated later in the text. 

K. Summary of Notation 

The notation introduced in this chapter comprises two nouns (_ and __) for the identity 
elements of minimum and maximum; two verbs take and drop ({. }.) for truncating a 
list;  the  commute  adverb  ~  ;    the  conjunction  &  to  bond  nouns  to  dyads;  and  verbs 
produced by the atop conjunction @ have dyadic as well as monadic cases.  

L. On Language 

Use the computer to test the following assertions: 

1.  The monad | yields the magnitude or absolute value. 

2.  The monad |. reverses its argument, and 3&|. rotates it by three places. 

  
 
 
 
 
 
3.  The monad -&| is equivalent to -@|, but the dyad -&| applies the dyad - to the 

result of applying the monad | to each argument. 

Chapter 2 Properties of Verbs  23 

4.  %&4 is division by 4, and is equivalent to 4&*^:_1 . 

5.  The monads +: and -: are double and halve. 

6.  The monads *: and %: are square and square root. 

7.   'abcde' is the list of the first five letters of the alphabet, and monads such as |. 

and 3&|. and 3 4&$ apply to it. 

Exercises 

A1  Define a verb sump that sums the positive elements of a list. 

Define dsq and sqd to double the square and square the double. 

Answer:  sump=:+/@(0&>.)  dsq=:(2&*)@(^&2) sqd=:^&2@(2&*) 

B1 Define the following verbs: 

from 

That subtracts its left argument from the right 

square 

Without using ^ 

double 

Without using * 

zero 

A monad that yields zero 

Answer:     from=: -~     square=:*~     double=:+~    zero=:-~  

C1  Test all the dyads defined thus far for associativity. 

D1  Which of the monads defined in preceding exercises are linear? 

E1  Use the arguments a=: 1 2 3 4 5 and b=: 3 1 5 2 4 to test 

all dyads (including -~ and ^~) for symmetry. 

E2  The expression ?~ n produces a random permutation of the  

integers i. n. Use it for further tests of symmetry. 

G1  Experiment with inverses of the monads defined in preceding 

exercises. 

H1  Test the dyad <. to see if (<./k{.a)<.(<./k}.a) agrees with 

<./a for various values of k and a . 

H2  Repeat Exercise H1 for the dyads - and ^  

H3  Characterize those dyads that satisfy the test of Exercise H1. 

Answer:    They are associative  

I1 

J1 

Experiment with various dyads to determine their identity elements. 

Experiment with the dyad % 

 
  
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
25 

Chapter 
3 

 Partitions and Selections 

A. Partition Adverbs 

The  partition  adverb  \  (called  prefix)  applies  to  monads  to  produce  many  useful  verbs. 
For example: 

   a=: 1 2 3 4 5 
   sum=: +/ 
   sum a 
15 

   sum\ a 
1 3 6 10 15 

Subtotals or “running” sums 

   (+/1),(+/1 2),(+/1 2 3),(+/1 2 3 4),(+/1 2 3 4 5) 
1 3 6 10 15 
   +/\a 
1 3 6 10 15 

Running products 

   */\a 
1 2 6 24 120 

   !a 
1 2 6 24 120 

   >./\ 3 1 4 1 5 9 
3 3 4 4 5 9 

Running maxima 

The partition adverb \. behaves similarly to produce a verb that applies to suffixes: 

   sum \.a 
15 14 12 9 5 

   */\.a 
120 120 60 20 5 

   <./\.3 1 4 1 5 9 

  
 
 
 
 
 
 
 
 
 
 
26  Arithmetic 

1 1 1 1 5 9 

   (*/\.a)*(*/\a) 
120 240 360 480 600 

  (+/\.a)+(+/\a) 
16 17 18 19 20 

   (-/\.a)-(-/\a) 
2 _1 2 1 2 

The diagonal adverb /. applies to (forward sloping) diagonals of tables. It will later be 
seen to be useful in multiplying polynomials and integers expressed in decimal. It is also 
useful in treating correlations and convolutions: 

   t=:1 2 1*/1 2 1 
   t 
1 2 1 
2 4 2 
1 2 1 

   sum/. t 
1 4 6 4 1 

   (sum/. t)*(10^i.-5) 
10000 4000 600 40 1 

   +/(sum/. t)*(10^i.-5) 
14641 

  121*121 
14641 

   +//.1 2 1*/1 3 3 1 
1 5 10 10 5 1 

   +//.1 3 3 1*/1 4 6 4 1 
1 7 21 35 35 21 7 1  

B. Selection Verbs 

The take and drop ({. and  }.) used in Section 2 H are examples of selection verbs. A 
more general selection is provided by the verb { (called from). For example: 

   primes=:2 3 5 7 11 13 
   2{primes 
5 

   0 2 4{primes 
2 5 11 

   3{.primes 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 3  Partitions And Selections  27 

2 3 5 
   (i.3){primes 
2 3 5 

   (i.-#primes){primes 
13 11 7 5 3 2 

   i.3 5 
 0  1  2  3  4 
 5  6  7  8  9 
10 11 12 13 14 

   0 2{i.3 5 
 0  1  2  3  4 
10 11 12 13 14 

   2 1 3 5 0 4{primes 
5 3 7 13 2 11 

The last sentence above is an example of a permutation that reorders the items of the list 
primes; a list such as 2 1 3 5 0 4 that produces a permutation is called a permutation 
list, or permutation vector, or simply a permutation. 

If the items of a list a are distinct, then the selection b=: i{a has an inverse in the sense 
that for a given b, an index can be found that selects it. The dyad i. fulfills this purpose, 
and is called indexing. For example: 

   a=:2 3 5 7 11 13 
   ]b=:3{a 
7 

   a i. b 
3 

   a i. 11 2 5 
4 0 2 

More precisely, the monads {&a and a&i. are mutually inverse. For example: 

   psel=: {&2 3 5 7 11 13 
   pind=: 2 3 5 7 11 13&i. 
   pind 7 2 
3 0 

   psel pind 7 2 
7 2 

A list such as a specifies a set of intervals, and an integer may be classified according to 
the interval in which it falls. More precisely, we will determine the index of the largest 
element  in  the  list  that  equals  or  precedes  it.  Thus,  5  and  6  both  lie  in  interval  2  of  a 
because they are greater than or equal to 2{a and less than 3{a. 

Indexing can be used to perform the classification as follows: 

 
  
 
 
 
 
 
 
 
 
 
 
 
28  Arithmetic 

   a 
2 3 5 7 11 13 

   x=: 6 
   x<a 
0 0 0 1 1 1 

   (x<a) i. 1 
3 

   ]i=: <:(x<a)i.1 
2 

   i{a 
5 

C. Grade and Sort 

The monad /: grades its argument. For example: 

   p=: 5 3 7 13 2 11 
   /:p 
4 1 0 2 5 3 

   (/:p){p 
2 3 5 7 11 13 

More precisely, the monad /: produces a permutation vector that can be used to sort its 
argument to ascending order. 

D. Residue 

Just  as  the  introduction  of  the  predecessor  as  the  inverse  of  the  successor  led  to  a  new 
class  of  numbers  outside  the  class  of  counting  numbers,  so  an  attempt  to  introduce  an 
inverse to a multiplication such as 5&* leads to new numbers when applied to an integer 
such as 17 that is not an integer multiple of 5. In other words, 17 is not in the (integer) 
domain of the inverse 5&*^:_1 . Similar remarks apply to an arbitrary multiple m&*. 

An  approximate  inverse  in  integers  can  be  obtained  by  locating  the  argument  in  the 
intervals specified by the multiples 5*i.n . For example: 

   x=: 17 
   m5=: 5*i.6 
   m5 
0 5 10 15 20 25 

   d=: <:(x<m5)i. 1 
   d 
3 

  5*d 
15 

   r=: x-5*d 

  
 
 
 
 
 
 
 
 
 
 
 
Chapter 3  Partitions And Selections  29 

   r 
2 
   5|x 
2 

The result r is the difference between the original argument and the nearest multiple of 5 
that does not exceed it; it is called the residue of x modulo 5, or the 5-residue of x . 

The dyad | is called residue, and x-m|x is an integer multiple of m. Consequently it is in 
the domain of the inverse m&*^:_1. Thus: 

   a=: i. 21 
   a 
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 
   8|a 
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 
   a-8|a 
0 0 0 0 0 0 0 0 8 8 8 8 8 8 8 8 16 16 16 16 16 

   8&*^:_1 a-8|a 
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 

   10&*^:_1 a-10|a 
0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 

E. Characters 

In English, the word Milk refers to a white liquid, whereas ‘Milk’ refers to the list of four 
literal characters ‘M’ and ‘i’ and ‘l’ and ‘k’. We will use quotes in a similar manner, as 
illustrated below: 

   alph=: ' ABCDEFGHIJKLMNOPQRSTUVWXYZ' 
   9 0 9 9 0 9 9 9 0 9 22 0 22 0 22 9 0 22 9 9 { alph 
I II III IV V VI VII 

t { ' *' 

   t=: 4>*/~ 3 2 1 0 1 2 3 
   t 
0 0 1 1 1 0 0 
0 0 1 1 1 0 0 
1 1 1 1 1 1 1 
1 1 1 1 1 1 1 
1 1 1 1 1 1 1 
0 0 1 1 1 0 0 
0 0 1 1 1 0 0 

*** 
*** 
******* 
******* 
******* 
*** 
*** 

   sentence=: '1 2 3^4' 
   reverse=: (i.-#sentence){sentence 
   reverse 
4^3 2 1    
   do=:". 
   do sentence 
1 16 81 
   do reverse 

 
  
 
 
 
 
 
 
30  Arithmetic 

64 16 4 

   ;: sentence 
+-----+-+-+ 
|1 2 3|^|4| 
+-----+-+-+ 
F. Box and Open 

The word-formation verb ;: can be applied to a character list that represents a sentence 
to break it into its individual words. Thus: 

   letters=: 'abc=:i.3 4+2' 
   words=: ;: letters 
   words 
+---+--+--+---+-+-+ 
|abc|=:|i.|3 4|+|2| 
+---+--+--+---+-+-+ 

   #words 
6 
   (i.-#words){words 
+-+-+---+--+--+---+ 
|2|+|3 4|i.|=:|abc| 
+-+-+---+--+--+---+ 

As illustrated, the result of the word-formation is a list of six items, each of which is a 
boxed list representing the corresponding word. 

A single box can also be formed by the box monad < as follows: 

   <'abcd' 
+----+ 
|abcd| 
+----+ 

   <2 3 5 
+-----+ 
|2 3 5| 
+-----+ 

   (<(<'abcd'),<2 3 5),<2 3$(<'abcd'),<2 3 5 
+------------+-------------------+ 
|            |+-----+-----+-----+| 
|+----+-----+||abcd |2 3 5|abcd || 
||abcd|2 3 5||+-----+-----+-----+| 
|+----+-----+||2 3 5|abcd |2 3 5|| 
|            |+-----+-----+-----+| 
+------------+-------------------+ 

The  box  verb    can  also  be  very  helpful  in  clarifying  the  behaviour  of  the  partition 
adverbs. For example:    

   <\a=:1 2 3 4 5 
+-+---+-----+-------+---------+ 
|1|1 2|1 2 3|1 2 3 4|1 2 3 4 5| 
+-+---+-----+-------+---------+ 

   <\.a 
+---------+-------+-----+---+-+ 

  
 
 
 
 
 
 
 
 
Chapter 3  Partitions And Selections  31 

|1 2 3 4 5|2 3 4 5|3 4 5|4 5|5| 
+---------+-------+-----+---+-+ 

   i. 3 4 
0 1  2  3 
4 5  6  7 
8 9 10 11 
   </.i.3 4 
+-+---+-----+-----+----+--+ 
|0|1 4|2 5 8|3 6 9|7 10|11| 
+-+---+-----+-----+----+--+ 

The monad > is the inverse of box; where necessary it “pads” the result with appropriate 
zeros or spaces. For example: 

   ]a=: ;: 'Gaily into Ruislip gardens' 
+-----+----+-------+-------+ 
|Gaily|into|Ruislip|gardens| 
+-----+----+-------+-------+ 
   >a 
Gaily   
into    
Ruislip 
gardens 

   b=:</.i.3 4 
   b 
+-+---+-----+-----+----+--+ 
|0|1 4|2 5 8|3 6 9|7 10|11| 
+-+---+-----+-----+----+--+ 

   >b 
 0  0 0 
 1  4 0 
 2  5 8 
 3  6 9 
 7 10 0 
11  0 0 

G. Summary of Notation 

The  notation  introduced  in  this  chapter  comprises  three  partition  adverbs,  prefix,  suffix, 
and oblique (\ \. /.); the dyads from and residue ({ |); and the monads box, open, 
grade, and word-formation (< > /: ;:). Section E also introduced the use of quotes to 
distinguish literals and other characters. 

H. On Language 

Review Section R of Chapter 1, and pursue one or more of the options suggested. 

In exercises first write (or at least sketch out) the result of each sentence without using 
the computer; then enter the sentence on the computer to check your answer. 

Exercises 

A1  q=:1 1&(*/) 

   q 1 2 1 

 
  
 
 
 
 
32  Arithmetic 

   r=:+//.@q 
   r 1 2 1 
   r 1 
   r r 1 
   r^:(5) 1 
   r^:(i.6 )  

A2     Experiment with the dyad ! for various cases, such as 3!5 and 4!5 and (i.6)!5. 

A3  (i.6)!5 

!/~i.6 

!~/~i.6 

   (!~/~i.6)=(r^:(i.6) 1) 

B1  (2*i.3){2 3 5 7 11 13 17 

   0 2 3 1{i.4 4 

   2{0 2 3 1{i.4 4 

B2  cl=:i.&1@< 

   6 cl 2 3 5 7 11 13 

   5 cl 2 3 5 7 11 13 

   4 cl 2 3 5 7 11 13 

B3  Experiment with negative left arguments to {. and }. and { 

D1  3|7 

   7|3 

   3|i.10 

   |/~i.7 

E1  text=:'i sing of olaf glad and big' 

   /: text 

   (/:text){text 

   text{~/:text 

   text/:text 

F1 

<\'abcdefg' 

   <\.'abcdefg' 

   a=:3 4$'abcde' 

   <\a 

<\.a 

  
 
33 

Chapter 
4 

 Representation of Integers 

A. Introduction 

Because  we  are  so  familiar  with  the  decimal  number  system  (which  extends 
systematically  to  larger  and  larger  numbers),  the  matter  of  distinct  representations  of 
successive  counting  numbers  did  not  pose  an  obvious  problem.  However,  in  a  system 
such  as  Roman  numerals,  the  sequence  I  II  III  IV  V  VI  VII  has  no  clear  pattern  of 
continuation beyond a few thousand. 

Although the decimal system is familiar, a careful examination of it is fruitful because it 
leads  to  simple  procedures  for  determining  the  results  of  verbs  such  as  addition, 
multiplication,  and  power.  We  begin  by  expressing  the  relationship  of  a  single  number 
(such as the number of days in a year) to the list of decimal digits that represent it: 

   n=:365 
   10^e 
100 10 1 
   d*10^e 
300 60 5 

d=:3 6 5 

e=:2 1 0 

+/d*10^e 

365 

The name e was chosen for the list 2 1 0 because the right argument of the power verb 
is often called an exponent. It could have been expressed using the verb i. as follows: 

   i. -3 
2 1 0 
   +/d*10^i.-3 
365 

The foregoing expression is, of course, suitable only for a list d of three items. To write a 
more general expression for any list d it is necessary to use a verb that yields the number 
of items of its list argument. Thus: 

   #d   
3 
   d=:1 7 7 6 
   +/d*10^i.-#d  

    +/d*10^i.-#d 

365 

  
 
 
 
 
 
34  Arithmetic 

1776 

The  foregoing  is  an  example  of  determining  the  base-10  value  of  a  list  of  digits,  and 
similar expressions apply for other number bases or radices. Thus: 

   +/d*8^i.-#d 
245 

   b=:1 1 0 1 
   +/b*2^i.-#b 
13 

   10#.d 
365 

   8#.d 
245 

   2#.b 
13 

The  last  three  sentences  show  the  use  of  the  dyad  #.  (called  base-value)  for  the  same 
evaluations. 

B. Addition 

Two lists representing numbers in decimal may be added to produce a representation of 
their sum, as illustrated below: 

   year=:3 6 5 
   agnes=: 3 0 4 
   base10=:10&#. 
   year + agnes 
6 6 9 

   base10 (year + agnes) 
669 

   (base10 year) + (base10 agnes) 
669 

   year+year 
6 12 10 

   base10 (year+year) 
730 

   (base10 year)+(base10 year) 
730 

Although  the  sum  year+year  yields  the  correct  sum  when  evaluated  by  base10,  it  is 
not in the usual normal form with each item in the list lying in the interval from 0 to 9. It 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 4  Representation of Integers  35 

can  be  brought  to  normal  form  by  subtracting  10  from  each  of  the  last  two  items  and 
“carrying” ones to the preceding items to obtain the result 7 3 0 in normal form. 

Since  a  zero  can  be  appended  to  the  beginning  of  a  list  without  changing  its  decimal 
value, lists of different lengths can be added by appending leading zeros to the shorter. 
For example: 

   dozen=:1 2 
   base10 0,dozen 
12 

   year+0,dozen 
3 7 7 

C. Multiplication 

A procedure for multiplication will first be stated, and its validity will then be examined: 

   a1=:3 6 5 
   b1=: 1 7 7 6 
   (base10 a1)*(base10 b1) 
648240 

   over=: ({.;}.)@":@, 
     by=: ' '&;@,.@[,.] 
   a1 by b1 over a1*/b1 
+-+----------+ 
| |1  7  7  6| 
+-+----------+ 
|3|3 21 21 18| 
|6|6 42 42 36| 
|5|5 35 35 30| 
+-+----------+ 

   a1*/b1 
3 21 21 18 
6 42 42 36 
5 35 35 30 

   ]p=:+//.a1*/b1 
3 27 68 95 71 30 

    base10 p 
648240 

Normalization of p by carries gives 6 4 8 2 4 0 and: 

   base10 6 4 8 2 4 0 
648240 

The foregoing procedure for multiplication comprises three steps: 

1. Form the multiplication table of the lists of digits. 

2. Sum the diagonals of the table. 

3. Normalize the sums. 

  
 
 
 
 
 
 
 
 
 
 
36  Arithmetic 

The  method  is  less  error-prone  than  the  one  commonly  taught,  which  distributes  the 
normalization  process  through  both  the  multiplication  and  summation  phases.  The 
validity of the process may be discerned from the following examples:  

   b1=:1 7 7 6 
   b2=:10^3 2 1 0 
   b=:b1*b2 
   b 
1000 700 70 6 

   a1=:3 6 5 
   a2=:10^2 1 0 
   a=:a1*a2 
   a 
300 60 5 

    (+/a)*(+/b) 
648240 

   a*/b 
300000 210000 21000 1800 
 60000  42000  4200  360 
  5000   3500   350   30 

   +/a*/b 
365000 255500 25550 2190 

   +/+/a*/b 
648240 

The  fact  that  the  product  of  the  sums  +/a  and  +/b  can  be  expressed  as  the  sum  of 
products arises from two properties: 

1. Multiplication distributes over addition. 

2. Summation (+/) is symmetric. 

In  the  expression  a*/b,  the  arguments  are  themselves  products  and,  because 
multiplication  is  both  associative  and  commutative,  a*/b  can  also  be  expressed  as  the 
product of two tables as follows: 

   a1*/b1 
3 21 21 18 
6 42 42 36 
5 35 35 30 

   a2*/b2 
100000 10000 1000 100 
 10000  1000  100  10 
  1000   100   10   1 

   (a1*/b1)*(a2*/b2) 
300000 210000 21000 1800 
 60000  42000  4200  360 
  5000   3500   350   30 

a*/b 

300000 210000 21000 1800 
 60000  42000  4200  360 
5000   3500   350   30 

Each element of the table  a1*/b1 is multiplied by the corresponding element from the 
“powers  of  ten”  table  a2*/b2,  and  those  elements  of  a1*/b1  multiplied  by  the  same 
power  of  ten  can  be  first  summed  and  then  multiplied  by  it.  Since  equal  powers  lie  on 

  
 
 
 
 
 
 
 
 
 
 
 
Chapter 4  Representation of Integers  37 

diagonals, 
p=:+//.a1*/b1 used in describing the multiplication procedure. 

the  sums  are  made  along 

these  diagonals,  as 

in 

the  expression 

The reason that equal powers lie on diagonals can be made clear by noting that a2 equals 
10^e=:2 1 0, that b2 equals 10^f=:3 2 1 0, and that a2*/b2 equals 10^e+/f : 

   e+/f 
5 4 3 2 
4 3 2 1 
3 2 1 0 

10^e+/f 

100000 10000 1000 100 
10000  1000  100  10 
1000   100   10   1 

D. Normalization 

The  normalization  process  used  in  Section  B  can  be  expressed  more  formally.  We  first 
define the main verbs to be used, and illustrate their use: 

   base10=:10&#. 
   residue=: 10&| 
   tithe=: 10&*^:_1 
   n=: 98 45 19 24 
   base10 n 
102714 

   remainder=: residue n 
   remainder 
8 5 9 4 

   n-remainder 
90 40 10 20 

   carry=: tithe n-remainder 
   carry 
9 4 1 2 

   carry ,: remainder    (,: laminates lists to form a table) 
9 4 1 2 
8 5 9 4 

   +//. carry ,: remainder 
9 12 6 11 4 

   base10 +//. carry ,: remainder 
102714 

We begin by specifying a “temporary” name t, and repeatedly re-assign to it the result of 
the process illustrated above: 

   t=: n 
   t=:+//. (tithe t-residue t) ,: residue t 

   t 
9 12 6 11 4 

   base10 t 
102714 

  
 
 
 
 
 
 
 
 
 
 
38  Arithmetic 

   t=:+//. (tithe t-residue t) ,: residue t 
   t   
0 10 2 7 1 4 

base10 t 

102714 

   t=:+//. (tithe t-residue t) ,: residue t 
   base10 t 
102714 

We will now use trains of isolated verbs (to be discussed below) to capture the foregoing 
process in a single verb, as follows: 

   reduce=: +//.@ ((tithe @ (] - residue)) ,: residue) 
   reduce n 
9 12 6 11 4 

   reduce ^:3 n 
0 1 0 2 7 1 4 

   reduce^:4 n 
0 0 1 0 2 7 1 4 

Because further repetitions of reduce continue to append leading zeros, we will instead 
use trim@reduce, where trim is defined to trim off a leading zero: 

   trim=:0&=@(0&{) }. ] 
   (trim @ reduce)^:3 n 
1 0 2 7 1 4 

  norm=: trim@reduce^:_ 

Three  repetitions  suffice  for  the  argument  n,  but  in  general  the  number  required  is 
unknown.  However,  since  the  process  v^:k  stops  when  the  successive  results  stop 
changing, it suffices to use a sufficiently large value of k, preferably infinity. 

We now consider the trains used in the definitions of reduce and trim. The phrase ] - 
residue occurring in the former has an obvious meaning, as illustrated below: 

   ] - residue n 
_8 _5 _9 _4 

However,  the  same  sequence  of  three  verbs  isolated  by  parentheses  (as  they  are  in  the 
definition of reduce) is called a train, and has the meaning illustrated below: 

   (] - residue) n 
90 40 10 20 
   (]n) - (residue n) 
90 40 10 20 

   (3&< <. 9&>) i. 15 
0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 4  Representation of Integers  39 

   (3&< i.15) <. (9&> i.15) 
0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 

Thus,  the  middle  verb  in  a  train  of  three  applies  dyadically  to  the  results  of  the  outer 
verbs. Such a train also has a dyadic meaning defined similarly. For example: 

   3 (+*-) 7 
_40 

   (3+7)*(3-7) 
_40 

   3 (< >. =) 2 3 4 5 
0 1 1 1 

  3<:2 3 4 5 
0 1 1 1 

E. Mixed Bases 

The base-value dyad #. used in Section A with the simple bases 10 and 8 and 2 can also 
be used with a mixed base defined by a list. For example: 

   base=: 7 24 60 60 
   base #. 0 1 2 3 

3723 

# of seconds in 0 days, 1 hour, 2 minutes, 3 seconds 

   a=:i. 2 4 
   a 
0 1 2 3 
4 5 6 7 
   base #. a 
3723 363967 

   base #: 3723 
0 1 2 3 

   base#: base #. a 
0 1 2 3 
4 5 6 7 

The last results illustrate the fact that the dyad #: provides an inverse to the base value, 
and can be used to produce the list representations of integers in any base. For example: 
   2 2 2 #: i. 8 
0 0 0 
0 0 1 
0 1 0 
0 1 1 
1 0 0 
1 0 1 

  
 
 
 
 
 
 
 
 
 
 
 
40  Arithmetic 

1 1 0 
1 1 1 

   10 10 10 #: 24 60 365 
0 2 4 
0 6 0 
3 6 5 

   fbase=: 3-i. 3 
   fbase 
3 2 1 
   fbase #: i.!3 
0 0 0 
0 1 0 
1 0 0 
1 1 0 
2 0 0 
2 1 0 

The final example employs an unusual “factorial” base, that will be used in the discussion 
of permutations in Chapter 7. 

F. Experimentation 

The verb mag=: ] >. - yields the magnitude of its argument; for example, mag 9 _9 
yields 9 9. However, the monad | does the same. 

Although  it  is  probably  unwise  to  spend  time  memorizing  bits  of  notation    before  they 
arise in context, it is worthwhile to experiment with the monadic cases of dyads already 
encountered  (and  conversely),  and  to  adopt  those  that  appear  useful.  The  language 
summary  at  the  back  of  the  book  can  be  used  to  suggest  further  experiments.  It  is  also 
worthwhile to experiment with the use of tables and other higher-rank arrays such as the 
rank-3 array i. 2 3 4 and the rank-4 array i. 2 3 4 5. Three matters merit attention: 

1.  Just  as  the  insertion  +/  inserts  the  verb  +  between  items  of  a  list,  so  does  it 
between items of a higher rank array: between the rows of a table, and between the 
planes  of  a  rank-3  array.  Consequently,  +/  applied  to  a  table  adds  one  row  to 
another. For example: 

   i. 3 4 
0 1  2  3 
4 5  6  7 
8 9 10 11 

+/i. 3 4 
12 15 18 21 

2.  Expressions such as a */ b, already used to form tables when applied to lists, 

also apply to higher-rank arrays. For example: 

   2 3 5 */ i. 2 4 
 0  2  4  6 
 8 10 12 14 

 0  3  6  9 
12 15 18 21 

  
 
 
 
 
 
 
Chapter 4  Representation of Integers  41 

 0  5 10 15 
20 25 30 35 
   1+i.2 3 
1 2 3 
4 5 6 

*// (1+i.2 3) 

4  5  6 
8 10 12 
12 15 18 

3.  The  rank  conjunction  "  determines  the  rank  of  the  sub-array  to  which  a  verb 

applies. For example: 

  sum=:+/ 
    ]a=:i. 2 3  
 0  1  2  3 
 4  5  6  7 
 8  9 10 11 

12 13 14 15 
16 17 18 19 
20 21 22 23 

   sum a                sum"2 a          sum"1 a 
12 14 16 18          12 15 18 21       6 22 38 
20 22 24 26          48 51 54 57      54 70 86 
28 30 32 34 

G. Summary of Notation 

Notation introduced in this chapter comprises 
isolated trains of verbs (as indicated in the 
diagrams at the right); one conjunction (rank ") ;             f   h    f   h 
|   |   / \ / \ 
and four verbs -- base value and its inverse, 
y   y   x y x y 
laminate, and magnitude (#. #: ,: |). 

        g        g 
 / \      / \ 

Exercises 

A1  base10=: 10&#. 
   base8=: 8&#. 
   base2=: 2&#. 
   a=:1 0 1 0 1 
   base2 a 
   base8 a 
   base10 a 

base2 -a 
base8 -a 
base10 -a 

C1  Compare  the  multiplication  process  described  at  the  beginning  of  Section  C  with 
the commonly-taught process for multiplying 365 by 1776 by actually performing 
both. 

C2  Repeat  Exercise  C1  for  various  arguments,  and  note  particularly  the  relative 

difficulties of reviewing the work for suspected errors. 

E1  What is the result of applying the verb norm to a single number such as 1776? 

  
  
 
 
 
 
 
 
 
 
42  Arithmetic 

E2  Enter t=: ?4 2$10 to define a table t of decimal digits. Then define a verb sum 
such that sum t gives the list representation of the integers represented by the rows 
of t. Check your result by applying base10 to it and +/base10 to t. 

Answer:  sum=: norm@(+/)  

E3  Write an expression that gives the list representation of the product of the integers 

represented by the rows of t. 

Answer:  norm +//."2^:(<:#t) *//t  

F1  Enter #: i. 8 and compare the result with the use of the dyad #: in Section E. 
Use further experiments to determine and state the definition of the monad #: . 

Answer:  #:x is equivalent to (n#2)#:x , where n is chosen just large enough to 
represent the largest element of x. 

F2  Define t=: ,"1~&0 , ,"1~&1 . Then enter ]b=:i.2 1 and t b and t t b, and 

so on, and compare the results with the results of #:i.2^k for various values of k . 

  
 
 
 
 
Chapter 
5 

 Proofs 

A. Introduction 

A proof is an exposition intended to convince a reader that a certain relation is true, and 
perhaps to provide some insight into why it is true. For example, Section O of Chapter 1 
provided, in passing, an illustration that the sum of the first six odd numbers was equal to 
six times six, that is, the square of six. Thus: 

   odds=:1+2*i. k=:6 
   odds 
1 3 5 7 9 11 

   +/odds 
36 

   k*k 
36 

   *:k 
36 

   *:#odds 
36 

This relation for the case of six odds suggests that a similar relation might hold for any 
number, and the prefix scan (\) provides a convenient test: 

   d=:1+i.15 
   d 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

   odds=:1+2*i.15 
   odds 
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 

   +/\odds 
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 

  
 
 
 
 
 
 
 
 
 
44 

   *:d 
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 

This result provides rather strong evidence that the sum +/1+2*i.k equals the square of 
k for any value of k, but it provides no insight into why this should be so. 

The  following  numbered  sequence  of  sentences  begins  and  ends  with  the  pair  whose 
equivalence  is  to  be  established.  The  intermediate  sentences  differ  in  simple  ways  that 
can provide insight into why the relations would hold true for any value of k: 

S1 

odds=:1+2*i.k=:10 
odds 

1 3 5 7 9 11 13 15 17 19 

S2  

+/odds 

100 

S3 

S4 

S5 

S6 

S7 

S8 

S9 

|.odds 

19 17 15 13 11 9 7 5 3 1 

+/|.odds 

100 

-: (+/odds) + (+/|.odds)  (-: halves its argument) 

100 

-: +/ (odds+|.odds) 
100 

+/ -: (odds+|.odds) 

100 

odds+|.odds 

20 20 20 20 20 20 20 20 20 20 

-: odds+|.odds 

10 10 10 10 10 10 10 10 10 10 

S10 

k#k 

10 10 10 10 10 10 10 10 10 10 

S11 

+/k#k 

S12 

S13 

100 

k*k 

100 

*:k 

100 

Sentences  S2  and  S4  to  S7  show  that  the  sum  of  the  first  ten  odds  can  be  written  in 
several  equivalent  ways,  but  really  demonstrate  it  only  for  the  specific  case  of  k=:10. 

 
 
 
 
    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
45 

However,  we  may  see  reasons  to  believe  that  the  relations between successive 
sentences should hold for other values of k. 

For  example,    because  +/  is  symmetric  (as  defined  in  Section  2  E),  and  because  the 
monad |. permutes its argument, S2 and S4 agree for any list odds . Further, in S5, one-
half of the sum of two equal things is equal to either one of them, and similarly simple 
arguments  can  establish  the  equality  of  the  pairs  S6,  S7;  S7,  S11;  S11,  S12;  and  S12, 
S13. In particular, S12 agrees with S11 because their agreement expresses the definition 
of multiplication. 

We will call a sequence such as S1-S13 an informal proof; it provides insight but leaves 
to the reader the task of providing precise reasons for the equivalence of certain pairs of 
sentences. A formal proof is one in which each sentence is annotated by a clear statement 
of the reasons for its equivalence with an earlier sentence. 

An  informal  proof  is  satisfactory  only  if  the  relations  between  successive  sentences  are 
obvious  to  the  reader.  If  so,  why  is  it  ever  desirable  to  make  formal  a  good  informal 
proof?  Firstly,  what  is  obvious  to  one  reader  may  not  be  to  another.  A  second,  more 
serious,  reason  is  that  obvious  reasons  for  relations  may,  in  fact,  be  wrong,  or  at  least 
incomplete. 

For example, does +/1+2*i.k equal k*k for the case k=:0 ? The answer is yes, but this 
does  not  follow  from  the  arguments  given  thus  far,  since  they  took  no  account  of  the 
definition  of  the  summation  of  an  empty  list.  A  complete  proof  would  require 
examination of the definition of identity elements in Section 2 I. 

In  the  foregoing  example  the  conclusion  remained  correct  even  though  the  reasons 
provided were incomplete, but unexamined proofs and definitions can also lead to errors 
or contradictions. For example, the prime numbers illustrated in Exercise O1 of Chapter 1 
have the important property that any counting number greater than one can be expressed 
as  a  product  of  one  or  more  primes,  and  that  this  factorization  is  unique.  For  example, 
using the first five elements of the list obtained in the cited exercise: 

   pr=:2 3 5 7 11 
   e=:2 0 2 1 0 
   pr^e 
4 1 25 7 1 
   */pr^e 
700 

Thus, the exponents 2 0 2 1 0 specify the prime factorization of the integer 700, and 
no other factorization in primes is possible. 

We turn now to a definition of primes that is commonly used in high-school: A prime is 
an integer that is divisible only by itself and one. The integers in the list pr satisfy this 
condition, but so does the integer 1. We now consider a list of “primes” that includes 1, 
and see that the factorization of the integer 700 in terms of it is not unique: 

   p=:pr,1 
   p 
2 3 5 7 11 1 

   */p^2 0 2 1 0 0 
700 
   */p^2 0 2 1 0 3 

    
 
 
 
 
46 

700 

The loss of unique factorization clearly lies in a definition of primes that admits 1 as a 
member. We turn to an informal development of primes that leads to a suitable definition: 

   i=:>:i.8 

   i 
1 2 3 4 5 6 7 8 

   rem=: i|/i 
   rem 
0 0 0 0 0 0 0 0 
1 0 1 0 1 0 1 0 
1 2 0 1 2 0 1 2 
1 2 3 0 1 2 3 0 
1 2 3 4 0 1 2 3 
1 2 3 4 5 0 1 2 
1 2 3 4 5 6 0 1 
1 2 3 4 5 6 7 0 

   +/div 
1 2 2 3 2 4 2 4 

   2=+/div 
0 1 1 0 1 0 1 0 

   (2=+/div)#i 
2 3 5 7 

div=: 0= i|/i 
div 

1 1 1 1 1 1 1 1 
0 1 0 1 0 1 0 1 
0 0 1 0 0 1 0 0 
0 0 0 1 0 0 0 1 
0 0 0 0 1 0 0 0 
0 0 0 0 0 1 0 0 
0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 1 

The table rem is the table of remainders (or residues), and div is a divisibility table that 
identifies zero remainders. The sum +/div sums the columns of div to yield the number 
of divisors of each of the integers i, and the final sentence selects those integers that have 
exactly two distinct divisors. It furnishes a suitable definition: A prime is an integer that 
has exactly two distinct divisors. 

We conclude this section with an example of an informal development designed to clarify 
some matters of elementary algebra.  

The  expression  a3  is  commonly  used  to  denote  what  we  denote  here  by  a^3,  and  is 
defined  by  saying  that  it  is  the  product  of  three  factors  a  (which  we  would  write  as 
a*a*a)  but  also  by  continuing  to  define  a0  as  1.  What  is  meant  by  a  product  of  no 
factors, and why should its result be 1 ? Somewhat less mysteriously, what is a product of 
one factor (a1), and why should it yield a ? 

The definitions of expressions such as a^n and !n are commonly extended to arguments 
that  do  not fall under the initial definition, by extending them so as to maintain certain 
significant “patterns” or “identities”. These patterns can often be made clear by applying 
functions to lists (such as i.n) that themselves maintain simple patterns. For example: 

   a=:4 

   e=:3 4 5 

   a^e 

 
 
 
 
 
 
 
 
 
 
 
47 

64 256 1024 

To  evaluate  the  next  in  sequence  (that  is,  a^6),  one  might  perform  the  calculation 
4*4*4*4*4*4  or,  more  efficiently,  note  that  the  result  is  simply  4 times the preceding 
case  a^5.  In  other  words,  the  pattern  extends  to  the  right  by  multiplication  by  4. 
Consequently, and more interestingly, it proceeds to the left by division by 4. Thus, since 
4^3 is 64, it follows that 4^2 is 16, that 4^1 is 4, and that 4^0 is 1.  

These last two results provide some insight into why a^1 and a^0 are defined as a and 1 
for any a, including the case where a itself is zero. It is worth noting that some college 
texts  state  that  0^0  is  undefined,  even  though  the  result  1  is  clearly  needed  to  make  it 
possible to evaluate the general form of the polynomial in x with coefficients c, namely, 
+/c*x^i.#c. 

Going,  for  a  moment,  outside  the  domain  of  the  integers,  we  may  also  note  that  the 
pattern continues through negative and fractional values. Thus: 

   a=:4 
   e=:3 4 5 
   a^e 
64 256 1024 

   e=:3-~i.7 
   e 
_3 _2 _1 0 1 2 3 

   4^e 
0.015625 0.0625 0.25 1 4 16 64 

   f=:-:i.6 
   f 
0 0.5 1 1.5 2 2.5 

   4^f 
1 2 4 8 16 32 

In the final example, there are two steps rather than one between successive integers of 
the equally-spaced elements of the exponent f, and 4^f must therefore exhibit a pattern 
of multiplication by a factor which applied twice produces multiplication by 4; in other 
words, a factor that is the square root of 4. 

B. Formal and Informal Proofs 

Although topics in mathematics are often presented deductively, as a sequence of formal 
proofs that appear to lead to collections of indisputable facts, we will continue to use an 
informal approach that emphasizes the use of expressions (such as the pair +/\odds and 
*:d of Section A) that suggest relations, and sequences of expressions (such as S1-S13) 
that outline a proof. 

The  reasons  for  adopting  such  an  informal  approach  are  rooted  mainly  in  the  view  of 
mathematics expressed clearly and entertainingly in the dialogue in Lakatos’ Proofs and 
Refutations  [5]  (discussed  briefly  in  Section  C),  but  also  in  the  characteristics  of  the 

    
 
 
 
 
 
 
48 

notation used here; characteristics that make it easy to express patterns in lists and tables, 
and  to  display  them  accurately  and  effortlessly  by  entering  the  expressions  on  a 
computer. 

To  appreciate  these  characteristics  the  reader  should  attempt  to  render  various 
expressions  in  this  text  clearly  and  completely  in  more  conventional  notation.  For 
example,  +/odds  may  be  expressed  by  using  sigma  notation,  but  +/\odds  would 
probably be expressed as: 

        i 
  ci = Σ oddsi   
       j=1 

an expression that does not yield an entire list as does +/\odds, but specifies it indirectly 
by specifying each of the elements of some list denoted by c. 

In  a  similar  vein,  it  might  be  assumed  that  the  sigma  notation  used  for  +/odds  would 
also serve for +/|.odds as follows: 

n 
Σ   oddsi 

       i=1 

1 
Σ  oddsi 
i=n 

However,  the  summation  from  n  to  1  is  normally  taken  to  denote  summation  over  an 
empty set, since no summation from j to k could otherwise denote the empty case. 

It might also be noted that the symbol n commonly used in sigma notation has no clear 
connection  to  the  number  of  elements  in  the  argument,  and  cannot  be  expressed  as  a 
function of the argument without introducing some notation analogous to #odds. 

C. Proofs and Refutations 

Of his Proofs and Refutations [4], Lakatos says “Its modest aim is to elaborate the point 
that  informal,  quasi-empirical,  mathematics  does  not  grow  through  the  monotonous 
increase  of  the  number  of  indubitably  established  theorems  but  through  the  incessant 
improvement  of  guesses  by  speculation  and  criticism,  by  the  logic  of  proofs  and 
refutations.” 

He  goes  on  to  say  that  there  is  a  simple  pattern  of  mathematical  discovery  -  or  of  the 
growth  of  informal  mathematical  theories  -  that  consists  of  the  following  stages  (also 
quoted from [4]): 

1.  Primitive conjecture 

2.  Proof  (a  rough  thought-experiment  or  argument,  decomposing  the  primitive 

conjecture into sub-conjectures or lemmas). 

3. 

‘Global’ counterexamples (counterexamples to the primitive conjecture) emerge. 

4.  Proof re-examined: the ‘guilty lemma’ to which the global counter-example is a 
‘local’  counterexample  is  spotted.  This  ‘guilty’  lemma  may  have  previously 
remained ‘hidden’ or may have been misidentified. Now it is made explicit, and 
built  into  the  primitive  conjecture  as  a  condition.  The  theorem  -  the  improved 
conjecture  -  supersedes  the  primitive  conjecture  with  the  new  proof-generated 
concept as its paramount new feature. 

 
 
 
 
 
49 

As  a  result,  “Counterexamples  are  turned  into  new  examples  -  new  fields  of 
inquiry open up.” 

Lakatos illustrates this process by following a simple conjecture through surprising twists 
and  turns,  citing  positions  held  by  dozens  of  eminent  mathematicians.  To  quote  from  a 
review  cited  on  the  cover,  “The  whole  book,  as  well  as  being  a  delightful  read,  is  of 
immense value to anyone concerned with mathematical education at any level.” 

We will illustrate the process briefly. Having counted the number of vertices v, edges e, 
and faces f of various polyhedra (bounded by multiple flat faces, surfaces, or “seats” as 
suggested by the root hedra), a class arrives at the conjecture that the expression f+v-e 
yields 2 for any polyhedron. For example: 

Tetrahedron 

Square-base pyramid 

Cube 

f 

4 

5 

6 

v 

4 

5 

8 

e 

6 

8 

12 

f+v-e 

2 

2 

2 

The  teacher  provides  the  following  proof  or  “thought-experiment”  to  establish  the 
validity of the relation for all polyhedra: 

1.  Triangulate  each  face  by  (repeatedly)  drawing  a  line  between  some  pair  of 
vertices not already joined by an edge. [In the square-based pyramid this requires 
one  diagonal  on  the  base;  in  the  cube  it  requires  one  diagonal  on  each  face.] 
Since each line drawn adds one edge and one face (splitting one existing face into 
two), the triangulation does not change the result of f+v-e. 

2.  Remove one face, leaving a hole bounded by three edges. 

3.  Dismantle the body triangle-by-triangle until only one remains, removing at each 
step one edge and one face, or one vertex, two edges, and one face. Either action 
leaves f+v-e unchanged. 

4.  For the final triangle, f+v-e is 1+3-3 (that is, 1), which, together with the face 

removed in step 2, gives a result of 2 for f+v-e. 

The validity of each step of the process is challenged by students who enter the dialogue, 
and the validity of the conjecture itself is challenged by counterexamples, including one 
provided by a body formed by fitting together into a square “picture frame” four identical 
moldings (polyhedra) having the following end and side views: 

    __          __________________________ 
   /  \     
  /    \       /                            \ 

 /                          \ 

A direct count gives 16+16-32 or 0, contradicting the conjecture. 

Attempts  are  first  made  to  sharpen  the  definition  of  a  polyhedron  so  as  to  save  the 
conjecture by barring the picture frame from consideration (as a “monster”), and later to 
revise the conjecture so as to account for such a monster. 

One such revision is based on the observation that the “well-behaved” polyhedra shared 
the property that (if constructed of elastic surfaces) they could be inflated to a sphere, but 
the picture frame could not. Moreover, a single cut through one limb of the frame (which 

    
 
 
 
50 

would appear as a vertical line in the side view above) would form a body with two new 
faces, eight new vertices, and eight new edges, restoring the result of 2 for f+v-e, and 
producing a body that could be inflated to a sphere. 

A revised conjecture taking into account the “connectedness” or “number of cuts needed 
to  produce  a  ‘spherical’  body”  can  therefore  be  formulated;  but  it  again  is  subject  to 
further criticism and refinement. 

We conclude this section with an extended quotation from Lakatos (page 73): 

TEACHER:  No! Facts do not suggest conjectures and do not support them either! 

BETA: 

Then  what  suggested  2=f+v-e  to  me  if  not  the  facts,  listed  in  my 
table? 

TEACHER: 

I  shall  tell  you.  You  yourself  said  you  failed  many  times  to  fit  them 
into  a  formula.  Now  what  happened  was  this:  you  had  three  or  four 
conjectures which in turn were quickly refuted. Your table was built up 
in the process of testing and refuting these conjectures. These dead and 
now  forgotten  conjectures  suggested  the  facts,  not  the  facts  the 
conjectures. Naive conjectures are not inductive conjectures: we arrive 
at them by trial and error, through conjectures and refutations. But if 
you - wrongly - believe that you arrived at them inductively, from your 
tables, if you believe that the longer the table, the more conjectures it 
will  suggest,  and  later  support,  you  may  waste  your  time  compiling 
unnecessary data. Also, being indoctrinated that the path of discovery 
is from facts to conjecture, and from conjecture to proof (the myth of 
induction),  you  may  completely  forget  about  the heuristic alternative: 
deductive guessing. 

D. Proofs 

Throughout this text we will present examples intended to stimulate the formulation of 
conjectures, but will not develop proofs. However, the reader is encouraged to provide 
formal  and  informal  proofs  for  any  conjectures  that  suggest  themselves.  The  present 
section  will  provide  examples  of  proofs  of  identities  that  are  familiar  in  elementary 
mathematics, but are often treated in more limited forms. 

In this section we will use the name X to denote a single element (or scalar), and other 
names to denote lists (or vectors). We will write one sentence below another to indicate 
that they are equivalent. Thus: 

Thm1: 

+/X*W 

X*+/W 

asserts  that  the  sum  over  a  scalar  times  a  list  is  equivalent  to the scalar times the sum 
over the list, and labels the identity as Thm1 (Theorem 1) for future reference. 

A formal proof of a theorem is provided by annotating each sentence after the first with 
the reason that it is equivalent to the sentence preceding it. Thus: 

Thm1: 

+/X*W 

    X*+/W 

X&* distributes over +   (Section 2 D) 

If  values  are  assigned  to  the  names  used  in  a  theorem,  then  each  sentence  may  be 
entered and executed as a test for the case of the particular values assigned. Thus: 

 
 
 
51 

   X=: 3 
   W=: 3 1 4 1 
   +/X*W 
27 

   X*+/W 
27 

This  executability  is  reassuring  in  developing  an  identity  or  proof,  because  a  mis-
statement will very likely produce a different result. For example: 

Thm2: V=: 2 4 6 

+/V*/W 
  36 12 48 12 

(+/V)*W 
  36 12 48 12 

Thm1 applied for each element of W 
(since +/V is a scalar) 

A  sequence  of  equivalent  sentences  implies  that  the  first  sentence  is  equivalent  to  the 
last.  Hence  the  following  is  a  formal  proof  that  the  sum  of  the  column  sums  of  the 
multiplication table V*/W equals the product of the sums +/V and +/W: 

Thm3: +/+/V*/W 

    +/V*(+/W) 

 Thm2 and commutativity of * 

    (+/V)*(+/W) 

Thm1 (with +/W for X and V for W) 
and commutativity of *. 

The following theorem can be proved formally by showing that the element of column j 
of row i of the first table is equal to the corresponding element of the second table: 
Thm4:  (A*P)*/(B*Q) 
      (A*/B)*(P*/Q) 

It can be illustrated as follows: 

   A=:2 3 5 
   B=: 3 1 4 1 
   P=: 4 3 2 
   Q=: 2 7 1 8 

   (A*P)*/(B*Q) 
48 56 32 64 
54 63 36 72 
60 70 40 80 

   (A*/B)*(P*/Q) 
48 56 32 64 
54 63 36 72 
60 70 40 80 

    
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
52 

Since  x^n is defined by */n#x, it is easy to show that (x^n)*(x^m) is equivalent to 
x^(m+n). This result can be used in the proof of the following theorem: 

Thm5:  (X^A)*/(X^B) 

      X^(A+/B) 

The foregoing theorems will be used in an exercise in Section B of Chapter 9 to prove 
that  the  product  of  two  polynomials  with  coefficients  C  and  D  is  equivalent  to  a 
polynomial with coefficients +//.C*/D. 

The fact that multiplication distributes over addition is commonly extended to a product 
of sums and expressed in conventional notation as: 

LHS= (a+A)(b+B) 

RHS= (ab)+(aB)+(Ab)+(AB) 

the left-hand side LHS being equivalent to the right-hand side RHS. 

This identity can be extended to a product over any number of sums as follows: 

LHS=(a+A)(b+B)(c+C) 

RHS=(abc)+(abC)+(aBc)+(aBC)+(Abc)+(AbC)+(ABc)+(ABC) 

LHS=(a+A)(b+B) ... (z+Z) 

The last expression above uses the informal three-dot notation to suggest continuation of 
the  same  form  to  arbitrary  lengths.  To  appreciate  the  difficulties  of  such  informal 
notation,  the  reader  should  attempt  its  use  in  a  clear  definition  of  the  corresponding 
right-hand side. 

The use of vectors (lists) makes the expression of the left-hand side simple: */v1+v2 , 
where (in the three-element case above), v1=:a,b,c and v2=:A,B,C. 

To clarify the pattern of the right-hand side, we will use explicit values for v1 and v2, 
thus allowing the direct evaluation of every expression. We will also use numbers less 
than ten in v1, and greater than ten in v2 to make patterns easier to recognize. Thus: 

   v1=:2 3 4 

    v2=:12 13 14 

v1+v2 
14 16 18 

   ]LHS=: */v1+v2 
4032 

   ]RHS=:(2*3*4)+(2*3*14)+(2*13*4)+(2*13*14)+(12*3*4)+ 
              (12*3*14)+(12*13*4)+(12*13*14) 
4032 

The pattern in the expression for RHS can be better seen in the following table: 

   M=:>2 3 4;2 3 14;2 13 4;2 13 14;12 3 4;12 3 14; 
              12 13 4;12 13 14 

 
 
 
   
 
 
 
 
 
53 

   M 
 2  3  4 
 2  3 14 
 2 13  4 
 2 13 14 
12  3  4 
12  3 14 
12 13  4 
12 13 14 

   */"1 M 
24 84 104 364 144 504 624 2184 

   +/*/"1 M 
4032 

Because  the  items  of  v2  exceed  10,  the  pattern  in  M  can  be  displayed  more  clearly  as 
booleans: 

   ]b1=: M<10 
1 1 1 
1 1 0 
1 0 1 
1 0 0 
0 1 1 
0 1 0 
0 0 1 
0 0 0 

]b2=: M>10 

0 0 0 
0 0 1 
0 1 0 
0 1 1 
1 0 0 
1 0 1 
1 1 0 
1 1 1 

The right-hand side can now be expressed in either of two ways: 

   ]RHS=: +/(*/"1 v1^b1)*(*/"1 v2^b2) 
4032 

   ]RHS=: +/*/"1 (v1,v2)^(b1,.b2) 
4032 

The  details  of  these  expressions  can  be  explored  by  displaying  the  partial  results.  For 
example, the rows of v1^b1 contain the appropriate elements from v1 with the elements 
from  v2  being  replaced  by  ones  (the  identity  element  of  *),  and  the  product  over  the 
rows  multiplied  by  the  product  over  the  rows  of  v2^b2  yields  the  products  to  be 
summed. Thus: 

   v1^b1 
2 3 4 
2 3 1 
2 1 4  
2 1 1  
1 3 4  
1 3 1  
1 1 4  

v2^b2 
  1  1  1 
  1  1 14 
  1 13  1 
  1 13 14 
 12  1  1 
 12  1 14 
 12 13  1 

    
 
 
 
 
 
 
 
 
54 

1 1 1  

 12 13 14 

   */"1 v1^b1 
24 6 8 2 12 3 4 1 
   */"1 v2^b2 
1 14 13 182 12 168 156 2184 

   (*/"1 v1^b1)*(*/"1 v2^b2) 
24 84 104 364 144 504 624 2184 

   +/(*/"1 v1^b1)*(*/"1 v2^b2) 
4032 

Comparison of b2 with the result of #:i.2^3 in Exercise F1 of Chapter 4 should make 
it clear that #:i.2^n is the table appropriate to any list v of n elements. Moreover, as 
illustrated 
the  verb  t=:  ,"1~&0,  ,"1~&1 
applied to #:i.2^n yields the table for a list of one more element. 

in  Exercise  F2  of  Chapter  4, 

The  foregoing  facts  can  be  used  to  formalize  the  following  proof  of  the  equality  of 
general  functions  for  the  results  illustrated  above  for  LHS  and  RHS.  We  first  define  the 
functions: 

   lhs=:*/@(+"1) 

   rhs=:+/@(f*g) 

     g=:*/"1@(]^T)@] 

     f=:*/"1@(]^0&=@T)@[ 

       T=: #:@i.@(2&^)@# 

For lists V and W of one element each, the results of V lhs W and V rhs W can easily 
be shown to be equivalent. We now present an inductive proof in which we assume that 
V  lhs  W  and  V  rhs  W  are  equivalent  for  any  lists  of  n  elements,  and  then  use  that 
induction hypothesis to prove that they are equivalent for lists on n+1 elements. Thus: 

(x,V) rhs (y,W) 

+/(x,V) (f*g) (y,W) 

+/(L=:(x,V)f(y,W))*(x,V)g(y,W) 

+/L**/"1(y,W)^T (y,W) 

Def of rhs 

Def of fork 

Def of g 

+/L**/"1(y,W)^(0,"1 U),(1,"1 U=:T W)   

Structure of T 

+/L*((y^0)*Q),(y^1)*Q=:*/"1 W^U   

+/L*Q,y*Q 

+/((x*P),P=:*/"1 V^0=U)*Q,y*Q 

+/(x*P*Q),y*P*Q      

(x+y)*+/P*Q 

(x+y)*V lhs W                                  

(x+y)**/V+W 

*/(x,V)+(y,W) 

Analogous 

treatment of L 

Induction 

hypothesis 

 
 
 
 
 
55 

(x,V) lhs (y,W) 

    
 
57 

Chapter 
6 

Logic 

A. Domain and Range 

As stated in Section 1 D, the domain of a verb is the collection of arguments to which it 
can apply. For example, the integer 4 is in the domain of >:, but the characters '!' and 
'b'  and '4' are not. 

Similarly, the range of a verb is the collection of results that it can produce. The verb >: 
can  produce  any  integer,  and  so  its  range  is  the  same  as  its  domain.  This  agreement  of 
range and domain also holds for verbs such as + and *; but not for %, which can produce 
fractions or rational numbers, and so has a wider range as discussed in Chapter 9. 

A verb may also have a range more limited than its domain. For example, the verb 4&| 
applies to any integer, but its results all fall in the finite list i.4, that is,0 1 2 3. 

It  is  sometimes  useful  to  examine  the  properties  of  a  verb  when  it  is  applied  only  to  a 
restricted part of its domain, particularly if it is restricted to its range. For example, when 
restricted to the domain i.4, the verbs: 

pm4=: 4&|@* 
sm4=: 4&|@+ 

(Product modulo 4) 
(Sum modulo 4) 

have the following tables: 

   pm4/~ i.4 
0 0 0 0 
0 1 2 3 
0 2 0 2 
0 3 2 1 

sm4/~ i.4 

0 1 2 3 
1 2 3 0 
2 3 0 1 
3 0 1 2 

We will use the phrase “v on d” to refer to the verb resulting from restricting the verb v 
to the domain d. For example, “4&|@* on i.4” refers to the product mod 4 restricted to 
the  domain  0  1  2  3,  and  “*  on  i.2”  refers  to  the  boolean  and,  to  be  discussed  in 
Section C. 

  
 
 
 
 
 
 
58  Arithmetic 

B. Propositions 

A proposition or truth-function is any statement which can be judged to be either true or 
false, and is therefore a verb having a range of two elements. Following Boole (the father 
of  symbolic  logic),  we  will  denote  these  elements  by  1  (for  true)  and  0 (for false). For 
example: 

   p=: <&5 
   p 3 
1 

   p a=:i.8 
1 1 1 1 1 0 0 0 

(p a)#a 

0 1 2 3 4 

   2=+/0=|/~ a 
0 0 1 1 0 1 0 1 

   a#~2=+/0=|/~ a 
2 3 5 7 

C. Booleans 

The  nouns  0  and  1  (the  range  of  propositions)  are  called  booleans,  and  a  verb  whose 
domain and range are boolean is called a boolean function, or boolean. For example,  * 
limited to booleans might be called and; its table would appear as follows: 

   and=:* 
   and/~ b=:0 1 
0 0 
0 1 

   ]c=:i.8 
0 1 2 3 4 5 6 7 

   (>&2 c) and (<&5 c) 
0 0 0 1 1 0 0 0 

   (>&2 and <&5) c 
0 0 0 1 1 0 0 0 

   c #~ (>&2 and <&5) c 
3 4 
   (] #~ >&2 and <&5) c 
3 4 

The sentence  (>&2  and  <&5) is a “compound” proposition whose result is true if the 
proposition >&2 is true and the proposition <&5 is true. 

A verb or may be defined similarly: 

   or=: *@+ 
   or/~b 
0 1 

 
 
 
 
 
 
 
 
 
 
 
Chapter 6  Logic  59 

1 1 

   (=&7 c) or (<&5 c) 
1 1 1 1 1 0 0 1 

Note that the dyad + may produce non-boolean results, from which the monad * (called 
signum) produces booleans. Thus: 

   * _2 0 2 
_1 0 1 

+/~ b 

* +/~b 

0 1 
1 2 

0 1 
1 1 

The booleans and and or are exceedingly useful. For example: 

   dof10=: 0&=@(|&10) 
   dof10 c =: 1+i. 20 

1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 

   c#~dof10 c  

1 2 5 10 

   dof15=: 0&=@(|&15) 
   c#~dof15 c 

1 3 5 15 

Divisors of ten 

Divisors of fifteen 

   c#~ (dof10 and dof15) c 

1 5 

Common divisors of ten and fifteen 

   >./c#~ (dof10 and dof15) c 

5 

GCD of 10 and 15 

   10 15 |~/ c 
0 0 1 2 0 4 3 2 1 0 10 10 10 10 10 10 10 10 10 10 
0 1 0 3 0 3 1 7 6 5  4  3  2  1  0 15 15 15 15 15 

   0=10 15 |~/ c 
1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 
1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 

   and/0=10 15 |~/ c 
1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 

   c #~ and/0=10 15 |~/ c 
1 5 

   >./c #~ and/0=10 15 |~/ c 
5 

GCD of ten and fifteen 

  
 
 
 
  
 
 
 
 
 
 
 
 
 
 
60  Arithmetic 

The dyad +. is defined to yield the greatest common divisor of its arguments: 

   10 +. 15 
5 

+./ 10 15 

5 

The least common multiple is denoted by *. as illustrated below: 

   10 *. 15 
30 

(10*15) % 10+.15 

30 

D. Primitives 

Verbs  (such  as  *  and  +  and  *.  and  i.)  that  are  denoted  by  single  words  are  called 
primitives, to distinguish them from derived verbs produced by phrases such as that (*@+) 
used to define the boolean or in Section C. Since primitives and derived verbs are treated 
identically, this distinction is of little consequence except to the designer of a language, 
who must choose what primitives to provide. 

Should new primitives be added for such important cases as the boolean and and or? Not 
if primitives already exist that give the appropriate results when restricted to the boolean 
domain. The dyads <. and >. (min and max) might be tested for this purpose. Thus: 

   and=: * 
   or=: *@+ 
   b=: 0 1 
   <./~b 
0 0 
0 1 
   and/~b 
0 0 
0 1 

>./~b 

0 1 
1 1 

or/~b 

0 1 
1 1 

But do min and max provide the appropriate identity elements? The identity element for 
or should be 0, and for and should be 1, as illustrated below: 

   0 or b 
0 1 

1 and b 

0 1 

However, the identity elements of min and max are infinities. Thus: 

   <./i.0 
_ 

>./i.0 

__ 

Other  candidates  for  or  and  and  when  restricted  to  booleans  are  the  greatest  common 
divisor  (+.)  and  the  least  common  multiple  (*.)  introduced  in  the  preceding  section. 
Thus: 

   +./~b 
0 1 
1 1 

*./~b 

0 0 
0 1 

 
 
 
 
 
 
 
 
 
 
 
   +./i.0 
0 

*./i.0 

1 

Hereafter, these primitives will be used for or and and. It may be noted that Boole also 
represented  or  and  and  by  then-current  symbols  for  plus  and  times,  but  without  the 
appended dot used here to distinguish them from these verbs. 

Chapter 6  Logic  61 

E. Boolean Dyads 

Are  there  any  other  boolean  dyads  in  addition  to  *.  and  +.  (and  and  or)?  If  so,  how 
many? 

To  answer  these  questions  we  first  display  the  tables  for  *.  and  +.,  together  with  the 
ravel of each produced by the monad , : 

   *./~ b=:0 1 
0 0 
0 1 

   ,*./~b 
0 0 0 1 

+./~ b=:0 1 

0 1 
1 1 

,+./~b 

0 1 1 1 

We then observe that each table must contain four elements, each of which must belong 
to the range 0 1. Since each element may have either of two values, there are 2*2*2*2, 
or 2^4, or 16 possible tables which, when ravelled to form a four-element list, must agree 
with one of the columns in the following transposed table: 

   |:T 
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 
0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 
0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 

For example, columns 1 and 7 represent *. and +. : 

   1{"1 T 
0 0 0 1 

   and=: 1 b. 
   and/~ 0 1 
0 0 
0 1 

   and/i. 0 
1 

7{"1 T 

0 1 1 1 

or=: 7 b. 
or/~ 0 1 

0 1 
1 1 

or/i. 0 

0 

As illustrated in the foregoing, the adverb b. applies to any of the indices (0 to 15) of the 
table T to produce the corresponding boolean dyad. It may be noted that the base-2 value 
of any row yields its index; for example, 2#.7{T is 7. 

  
 
 
 
 
 
 
 
 
 
62  Arithmetic 

F. Boolean Monads 

A monad that negates a boolean argument is equivalent to subtraction from 1; it is called 
not, and is denoted by -. . There are in all four boolean monads as illustrated below: 

   b 
0 1 

   -. b 
1 0 

   ] b 
0 1 

   ~:~ b 
0 0 

   =~ b 
1 1 

G. Generators 

In English, compound propositions are commonly expressed using only or, and, and not. 
For example, using p, q, and r to denote propositions, and using parentheses to express 
the punctuation clearly: 

  p and q 

  not (p and q) 

  (p or q) and not (p and q) 

  not p and (not q) 

(p or q) or (p or not q) 

  (p and q) and (p and not q) 

(1 b.) 

(14 b.) 

(6 b.) 

(13 b.) 

(15 b.)    

(0 b.) 

Exclusive-or 

Implication 

True 

False 

Each of the foregoing phrases can be restated as definitions of verbs.  For example: 

   exclor=: +. *. -.@*. 
   exclor/~ 0 1 
0 1 
1 0 

Can all of the sixteen booleans be expressed using only or, and, and not ? The answer is 
yes, and for this reason the collection of verbs +. *. -. is said to be a set of generators 
of the booleans. For example, the case 0 b. (which yields 0 for every pair of arguments) 
can  be  expressed  as  (p  and  q)  and  (p  and  not  q),  and  15  b.  as  not 
(p and q) and (p and not q). 

 
 
 
 
 
 
 
 
 
 
 
Chapter 6  Logic  63 

Is +. *. -. a minimal set of generators, or could one of them be omitted? This is easily 
answered  by  showing  that  *.  itself  can  be  expressed  in  terms  of  +.  and  -.  and  can 
therefore be omitted: 

  and is not (not p) or (not q) 

The foregoing relation is sometimes expressed as “and is the dual of or (with respect to 
negation).” 

The  use  of  or  and  not  as  the  only  generators  can  lead  to  cumbersome  expressions  for 
some of the booleans, but all can be expressed in terms of them. 

Can a single boolean serve as generator? It can be shown that either 8 b. (not-or or nor) 
or 14 b. (not-and or nand) will serve. This matter is developed in exercises. 

H. Boolean Primitives 

The primitives +. and *. (gcd and lcm) when restricted to the boolean domain provide 
the  important  boolean  verbs  or  and  and.  Others  are  provided  by  similarly  restricting 
relations: 

< 

<: 

= 

>: 

> 

~: 

4 b. 

13 b. 

9 b. 

11 b. 

2 b. 

6 b. 

Implication 

Identity 

Exclusive-or 

Finally, +: and *: denote nor and nand, that is, 8 b. and 14 b. . 

I. Summary of Notation 

The  notation  introduced  in  this  chapter  comprises  one  adverb  boolean  (b.);  five  dyads 
or, and, nor, nand,  and not-equal  (+. *. +: *: ~:); three monads not, signum, and 
ravel (-. * ,). 

Exercises 

A1  Predict and test the results of n | (i. n) +/ (i. n) and of n | (i. n) */ 

(i. n) for various values of n including 10. 

A2  Define monads S and P such that S n and P n yield the tables of Exercise A1. 

Answer: 

 S=: ] | i. +/ i. and P=:]|i.*/i. 

B1  Predict  and  test  the  result  of  applying  to  an  integer  n  the  verb  PR=:  i.  #~ 
T@(+/)@(0&=)@(|/~)@i. for the cases T=:2&= and T=:2&< and T=:3&= . 

B2  Define  and  test  a  verb  IN  such  that  a  IN  b  yields  1  if  a  lies  in  the  interval 

between the smallest and largest elements of b. 

  
 
 
 
 
 
 
 
 
 
 
64  Arithmetic 

Answer: 

IN=: (<./@] < [)*.(>./@] > [) 

B3  Define  a  verb  L  such  that  a  L  b  lists  the  elements  of  a  that  lie  in  the  interval 

defined by b. 

Answer: 

L=: IN#[ 

C1  Explain the equivalence of the dyads *. and *%+. and test it in expressions such as 

(?7#100) (*. = * % +.)/ (? 10#100) . 

E1  The  verbs  1  b.  and  7  b.  may  be  called  and  and  or.  Recall  or  invent  suitable 
names for as many of the remaining fourteen boolean functions as you can. 

G1  Using  only  NAND=:  14  b.  define  a  monad  called  NOT  that  is  equivalent  to  the 

monad -. on the boolean domain. 

Answer: 

NOT=: NAND~ 

G2  Using only NAND=: 14 b.and NOT define dyads AND and OR that are equal to *. 

and +. on the boolean domain. 

Answer:  AND=: NOT@NAND       OR=:NOT@(NOT AND NOT) 

G3  Repeat Exercises G1, G2 using NOR=: 8 b. instead of NAND. 

 
 
 
 
 
 
65 

Chapter 
7 

Permutations 

A. Introduction 

Permute  is  a  verb  meaning  “to  change  the  order  of”,  and  |.  is  an  example  of  a 
permutation: 

   |. 'abcdef' 
fedcba 

   |. i. 5 
4 3 2 1 0 

Indexing provides arbitrary permutations. For example: 

   2 0 1 5 4 3 { 'abcdef' 
cabfed 

A  list  of  indices  to  {  that  produces  a  permutation  is  called  a  permutation  vector,  or 
permutation,  and  one  that  contains  n  elements  is  called  a  permutation  of  order  n.  A 
permutation of order n is itself a permutation of the list i. n. 

To  enumerate  all  permutations  of  order  n,  it  is  best  to  list  them  in  ascending  order 
(ascending  when  considered  as  the  digits  representing  an  integer),  as  illustrated  in  the 
following tables: 

p2 

0 1 
1 0 

p1 

0 

   p3 
0 1 2 
0 2 1 
1 0 2 
1 2 0 
2 0 1 
2 1 0 

   i=:i.!3 

  
 
 
 
 
 
 
 
 
 
66  Arithmetic 

   i{p4         (6+i){p4      (12+i){p4      (18+i){p4 
0 1 2 3      1 0 2 3       2 0 1 3        3 0 1 2 
0 1 3 2      1 0 3 2       2 0 3 1        3 0 2 1 
0 2 1 3      1 0 3 2       2 1 0 3        3 1 0 2 
0 2 3 1      1 2 3 0       2 1 3 0        3 1 2 0 
0 3 1 2      1 3 0 2       2 3 0 1        3 2 0 1 
0 3 2 1      1 3 2 0       2 3 1 0        3 2 1 0 

A  row  (or  rows)  of  any  one  of  these  tables  can  be  applied  to  index  (and  therefore  to 
permute) a list of the appropriate number of items. For example: 

   3{p4 
0 2 3 1 

   (3{p4){'abcd' 
acdb 

   (3 4{p4){'abcd' 
acdb 
adbc 

   (3 4{p4){i.4 
0 2 3 1 
0 3 1 2 

   p3{'abc' 
abc 
acb 
bac 
bca 
cab 
cba 

   3 A. 'abcd' 
acdb 

   3 4 A. 'abcd' 
acdb 
adbc 

p2{'ab' 

ab 
ba 

The  last  examples  illustrate  the  use  of  the  dyad  A.  in  which  i  A.  y  permutes  y  by  a 
permutation  of  order  #y,  the  permutation  being  row  i  of  the  corresponding  table of all 
permutations of that order. 

The index i in the phrase i A. y can be thought of as an atomic (that is, single-element) 
representation  of  the  permutation  vector  it  applies,  thus  providing  a  mnemonic  for  the 
word A. . 

From  these  examples  it  should  be  clear  that  the  phrase  (i.!n)A.i.n will produce the 
complete table of !n permutations of order n. Thus: 

 
 
 
 
 
 
 
 
 
 
 
 
   PT=: i.@! A. i. 

Chapter 7 Permutations  67 

PT 2 

0 1 
1 0 

PT 1 

0 

   PT 3 
0 1 2 
0 2 1 
1 0 2 
1 2 0 
2 0 1 
2 1 0 

B. Arrangements 

Any selection of k different items from a list is called an arrangement, or k-arrangement. 
For  example,  0  1{a  and  1  0{a  and  3  1{a  are  2-arrangements  from  the  list 
a=:'abcd'. 

Any k columns of a permutation table will contain all k-arrangements, each arrangement 
appearing !k times. For example: 

   CLAR2 
ab 
ac 
ad 
ba 
bc 
bd 
ca 
cb 
cd 
da 
db 
dc 

AR2 

   ALL=: (PT #a) { a=:'abcd' 
   AR2=: 2 {."1 ALL 
   CLAR2=: ~. AR2 
   ALL 
abcd 
abdc 
acbd 
acdb 
adbc 
adcb 
bacd 
badc 
bcad 
bcda 
bdac 
bdca 
cabd 
cadb 
cbad 
cbda 
cdab 
cdba 
dabc 
dacb 
dbac 
dbca 
dcab 
dcba 

ab 
ab 
ac 
ac 
ad 
ad 
ba 
ba 
bc 
bc 
bd 
bd 
ca 
ca 
cb 
cb 
cd 
cd 
da 
da 
db 
db 
dc 
dc 

The  table  ALL  contains  all  permutations  of  the  list  a;  the  table  AR2  contains  all  2-
arrangements,  with  each  arrangement  appearing  twice;  the  table  CLAR2  is  the  “clean” 
table  of  arrangements  with  redundant  items  suppressed.  The  suppression  of  redundant 
items is performed by the monad ~. (called nub).    

  
 
 
 
 
68  Arithmetic 

C. Combinations 

The arrangement 'ca' that occurs in the table CLAR2 is a permutation of the arrangement 
'ac', and the two cases therefore represent the same combination of elements from the 
list a=: 'abcd'. We may obtain a table of all 2-combinations of a by first sorting each 
row of CLAR2, and then taking the nub of the sorted table: 

~./:~"1 CLAR2 

ab 
ac 
ad 
bc 
bd 
cd 

   /:~"1 CLAR2 
ab 
ac 
ad 
ab 
bc 
bd 
ac 
bc 
cd 
ad 
bd 
cd 

The steps in the development of combinations can now be assembled to define a verb C 
such that k C n produces the table of all k-combinations of order n: 

   nub=: ~. 
   rtake=: {."1 
   rsort=: /:~"1 
   C=: nub@rsort@nub@([ rtake (PT@])) 
   2 C 4 
0 1 
0 2 
0 3 
1 2 
1 3 
2 3 

ab 
ac 
ad 
bc 
bd 
cd 

(2 C #a){a=: 'abcd' 

3 C 3 

0 1 2 

   1 C 3 
0 
1 
2 

   2 C 5 
0 1 
0 2 
0 3 
0 4 
1 2 
1 3 
1 4 
2 3 
2 4 

2 C 3 

0 1 
0 2 
1 2 

3 C 5 

0 1 2 
0 1 3 
0 1 4 
0 2 3 
0 2 4 
0 3 4 
1 2 3 
1 2 4 
1 3 4 

 
 
 
 
 
 
Chapter 7 Permutations  69 

3 4 

2 3 4 

   $ 2 C 5 
10 2 

$ 3 C 5 

10 3 

   (!5)%(!2)*(!5-2) 
10 

   (!5)%(!3)*(!5-3) 
10 

The  foregoing  definition  of  C  shows  clearly  the  relation  of  combinations  to  the 
permutations  of  the  corresponding  order.  However,  it  is  highly  inefficient  in  the  sense 
that k C n generates and sorts a large table (of r=:!n rows and n columns) in order to 
select from it a smaller table (of r%(!k)*(!n-k) rows and k columns). A more efficient 
alternative is developed in Exercise J10 of Chapter 9. 

As  illustrated  by  the  preceding  examples,  the  number  of  k-combinations  of  order  n  is 
given  by  (!n)%(!k)*(!n-k).  The  number  of  combinations  is  a  commonly-useful 
result; so important that the corresponding verb is treated as a primitive. For example: 

   2!5 
10 

(i.6)!5 
1 5 10 10 5 1 

   !/~i.6 
1 1 1 1 1  1 
0 1 2 3 4  5 
0 0 1 3 6 10 
0 0 0 1 4 10 
0 0 0 0 1  5 
0 0 0 0 0  1 

The last result is called the table of binomial coefficients; when transposed and displayed 
without the relevant sub-diagonal zeros it is also called Pascal’s triangle. 

D. Products of Permutations 

If p is a permutation vector, then the verb p&{ is a permutation. For example: 

   p=: 2 3 4 1 0 5 
   P=:p&{ 
   P a=:'abcdef' 
cdebaf 

   P^:2 a 
ebadcf 

P P a 

ebadcf 

   P^:0 1 2 3 4 5 6 7 8 a 
abcdef 
cdebaf 
ebadcf 
adcbef 
cbedaf 
edabcf 

P^:(i.9) i.6 

0 1 2 3 4 5 
2 3 4 1 0 5 
4 1 0 3 2 5 
0 3 2 1 4 5 
2 1 4 3 0 5 
4 3 0 1 2 5 

  
 
 
 
 
 
 
 
 
 
70  Arithmetic 

abcdef 
cdebaf 
ebadcf 

0 1 2 3 4 5 
2 3 4 1 0 5 
4 1 0 3 2 5 

In the foregoing it may be noted that the sixth power of the permutation P agrees with its 
original  argument,  and  the  pattern  therefore  repeats  thereafter.  The  period  of  this 
particular permutation is therefore said to be 6. 

E. Cycles 

Column 3 of the tables produced by the power of the permutation P of Section D shows 
that position 3 of successive powers is occupied by the elements 'd', and 'b' (or 3 1) 
in a repeating cycle of length two. Column 1 shows the same cycle displaced. 

Similarly,  column  4  shows  the  length-3  cycle  4  0  2,  and  columns  0  and  2  show  the 
same cycle displaced; column 5 shows the 1-cycle 5.  

The permutation P could therefore be represented unambiguously by its cycles as follows: 

   c=: 3 1 ; 4 0 2 ; 5 
   c 
+---+-----+-+ 
|3 1|4 0 2|5| 
+---+-----+-+ 
The dyad C. produces permutations specified in cycle form. Thus: 

   c C. a=:'abcdef' 
cdebaf 

   p { a 
cdebaf 

   p C. a 
cdebaf 

As  illustrated  by  the  last  example,  the  dyad  C.  also  accepts  permutation  vectors  as  the 
left  argument,  and  in  that  case  is  equivalent  to  the  dyad  {  .  Finally,  the  monad  C. 
provides  a  self-inverse  transformation  between  the  cycle  and  permutation-vector 
representations of a permutation. Thus: 
   C. c 
2 3 4 1 0 5 
   C. C. c 
+---+-----+-+ 
|3 1|4 0 2|5| 
+---+-----+-+ 
   PT=: i.@! A. i.    
   (PT 3);(C. PT 3);(C. C. PT 3) 
+-----+-------------+-----+ 
|     |+-----+---+-+|     | 
|     ||  0  | 1 |2||     | 
|     |+-----+---+-+|     | 
|0 1 2||  0  |2 1| ||0 1 2| 
|0 2 1|+-----+---+-+|0 2 1| 
|1 0 2|| 1 0 | 2 | ||1 0 2| 
|1 2 0|+-----+---+-+|1 2 0| 

 
 
 
 
 
Chapter 7 Permutations  71 

|2 0 1||2 0 1|   | ||2 0 1| 
|2 1 0|+-----+---+-+|2 1 0| 
|     ||2 1 0|   | ||     | 
|     |+-----+---+-+|     | 
|     ||  1  |2 0| ||     | 
|     |+-----+---+-+|     | 
+-----+-------------+-----+ 
From  columns  0  and  1  of  the  table  of  Section  D  it  may  be  seen  that  the  return  to  an 
identity permutation can occur only when the two cycles (of lengths 2 and 3) complete at 
the same time, in this case after  2*3 applications of the permutation. The period of the 
permutation is therefore 6. 

In general, the period of a permutation is the least common multiple of the lengths of its 
cycles. This will be illustrated further by a permutation of order 20 : 

   p20=:17 4 9 7 12 14 18 13 0 6 15 1 16 10 2 8 3 19 5 11 
   ]c20=:C. p20 
+-------------+-----------------------------------+ 
|18 5 14 2 9 6|19 11 1 4 12 16 3 7 13 10 15 8 0 17| 
+-------------+-----------------------------------+ 
   #@> c20 
6 14 
   p20&{^:18 a=: 'abcdefghijklmnopqrst' 
bdcphfgiljrqnaotkesm 

*./#@> c20 

42 

   p20&{^:(i.19) 'abcdefghijklmnopqrst' 
abcdefghijklmnopqrst 
rejhmosnagpbqkcidtfl 
tmgnqcfkrsiedpjahlob 
lqskdjoptfamhigrnbce 
bdfphgcilorqnastkejm 
ehoinsjabctdkrflpmgq 
mncakfgrejlhptobiqsd 
qkjrpostmgbnilceadfh 
dpgticflqsekabjmrhon 
hislajobdfmpregqtnck 
nafbrgcehoqitmsdlkjp 
kroetsjmncdalqfhbpgi 
ptcmlfgqkjhrbdoneisa 
iljqbosdpgntehckmafr 
abgdecfhisklmnjpqrot 
reshmjonafpbqkgidtcl 
tmfnqgckroiedpsahljb 
lqokdsjptcamhifrnbge 
bdcphfgiljrqnaotkesm 

F. Reduced Representation 

There are exactly !n permutations of order n, and the “factorial” base n-i.n introduced 
in  Section  4  E  can  be  seen  to  provide  exactly  !n  distinct  lists  of  n  integers,  each 
belonging to i.n: 

   R=: (]-i.) #: i.@! 
   R 3 
0 0 0 
0 1 0 

  
 
 
72  Arithmetic 

1 0 0 
1 1 0 
2 0 0 
2 1 0 

These  lists  can  be  used  to  represent  the  permutations  in  what  we  will  call  a  reduced 
representation, as distinguished from the “direct” representation used thus far: 

   D=: i.@! A. i. 
   D 3 
0 1 2 
0 2 1 
1 0 2 
1 2 0 
2 0 1 
2 1 0 

We will now define a verb RFD to yield the reduced representation from the direct, and an 
inverse DFR: 

   RFD=: +/@({.>}.)\."1 
   DFR=: /:^:2@,/"1 

For example: 

   RFD D 3 
0 0 0 
0 1 0 
1 0 0 
1 1 0 
2 0 0 
2 1 0 

DFR R 3 

0 1 2 
0 2 1 
1 0 2 
1 2 0 
2 0 1 
2 1 0 

The definitions of these verbs will be discussed in exercises. 

G. Summary of Notation 

The notation introduced in this chapter comprises five verbs: atomic permutation, cycle, 
nub, number of combinations, and random (A. C. ~. ! ?). 

Exercises 

A1  Using  as  argument  a  list  of  four  items,  test  the  assertion  that  the  monad  |.  is  a 
permutation, and determine the value of k such that k&A. is equivalent to |. . 

A2  Repeat Exercise A1 for the cases of lists of two, three, and five items. 

A3  Test  the  assertion  that  a  rotation  such  as  r&|.  is  a  permutation,  and  repeat 

Exercises A1 and A2 using rotations instead of reversal. 

A4  Apply the monad A. to various permutation vectors, and state its definition. 

 
 
 
 
 
 
 
 
A5  Experiment with k A. 'abcd' for negative values of k. 

B1  Write an expression for the number of k-arrangements of order n. 

C1  Define a monad  BC such that  BC  n gives the table of binomial coefficients up to 

Chapter 7 Permutations  73 

order n-1. 

Answer: 

  BC=: !/~@i. 

C2  Without using ! or BC define a monad CS that gives the column sums of BC n. 

Answer: 

CS=: 2&^@i.   

D1  Determine the power of the permutation p=: 4824 A. i. 7. 

Hint: 

Examine the table produced by p&{^:(i.20) i.7 

D2  Determine the power of the random permutation q=: 5?5. 

E1  Predict and test the results of C. k A. i.n for various values of k and n. 

E2  Predict and test the result of C. 1 3;2 0 4. 

E3  Repeat Exercise E2 for various boxed arguments of C. . 

E4  Use  various  permutations  p  to  test  the  assertion  that  the  power  of  p  is  the  least 

common multiple of the lengths of the cycles in its cycle representation. 

E5  Define a monad PER to give the power of a permutation p. 

Answer: 

PER=: *./@(#@>@C.) 

E6  What is the maximum period of a permutation of order n ? 

F1 

Predict  and  test  the  results  of  R  4  and  D  4  and  RFD  D  4  and  DFR  R  4  and 
(RFD@D = R) 4. 

F2  Define rfd equivalent to RFD except that it will apply only to a single permutation 

and not to a table of permutations. 

Answer: 

Omit "1 from RFD. 

F3  Analyze  the  definition  of  rfd  of  the  preceding  exercise  by  defining  and 

individually applying two functions such that f @ (g \.) is equivalent to rfd. 

Answer: 

f=:+/    g=: {.<}. 

F4  Analyze DFR. 

  
 
 
 
 
 
 
75 

Chapter 
8 

Classification and Sets 

A. Introduction 

It  is  often  necessary  to  separate  a  collection  of  objects  into  several  classes,  and  then 
perform some operation upon each of the classes. The operation performed is often trivial 
compared  to  the  complexity  of  the  classification  procedure  itself,  and  classification  is 
therefore  an  important  matter.  Indeed,  most  computation  involves  some  classification, 
even though the classification process may be implicit rather than explicit. 

As an example of the use of classification, consider a set of transactions that are recorded 
as a list of account numbers and a corresponding list of credits to the accounts. Thus: 

   an=: 1010 1040 1030 1030 1020 1010 1040 1040 1050 
   cr=:  131  755  458  532  218   47  678  679  934 

A summary should therefore post the sum 131+47 to account 1010  and 218 to account 
1020, and so on. If: 

   all=: 1010 1020 1030 1040 1050 

is the list of all account numbers, then c=: all =/ an is the classification table, and: 

   c=: all =/ an 
   c 
1 0 0 0 0 1 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 1 1 0 0 0 0 0 
0 1 0 0 0 0 1 1 0 
0 0 0 0 0 0 0 0 1 
   c*cr 
131   0   0   0   0 47   0   0   0 
  0   0   0   0 218  0   0   0   0 
  0   0 458 532   0  0   0   0   0 
  0 755   0   0   0  0 678 679   0 
  0   0   0   0   0  0   0   0 934 

   +/"1 c*cr 

  
 
 
 
 
 
 
76  Arithmetic 

178 218 990 2112 934 

The  classification  represented  by  the  table  c  is  both  complete  (each  element  being 
assigned  to  some  class)  and  disjoint  (each  element  being  assigned  to  no  more  than  one 
class). Classifications that arise from the expression a =/ b are disjoint if the elements 
of a are all distinct, and are complete if every element of b belongs to a. A boolean table 
B  represents  a  complete  disjoint  classification  if and only if each of its column sums is 
equal to 1; that is, if *./1=+/B . 

Since  a  table  provides  such  a  convenient  representation  of  a  classification,  we  will 
henceforth  speak  (rather  loosely)  of  the  table  itself  as  a  classification,  or  as  an  n-way 
classification, where n=:#B. 

Meaningful classifications need not be disjoint. For example, the letters of the alphabet 
may be classified phonetically by a 27-column table as follows: 

   a=:'abcdefghijklmnopqrstuvwxyz ' 
   PH 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 
1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 
0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 

   (0{PH)#a 
sz 
   a#~1{PH 
fv 

   a#~2{PH 
bdpt 
   a#~3{PH 
aeiouy 

Sibilants 

Fricatives 

Plosives 

Vowels 

   a#~4{PH 
bcdfghjklmnpqrstvwxz 

Consonants 

   a#~ >/4 2{PH 
cfghjklmnqrsvwxz 

Consonants that are not plosives 

Moreover, if t is any text, then (a i. t){"1 PH provides classifications of it: 

   t=: 'i sing of olaf' 
   a i. t 
8 26 18 8 13 6 26 14 5 26 14 11 0 5 

   (a i. t) {"1 PH 
0 0 1 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 0 0 0 0 1 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 
1 0 0 1 0 0 0 1 0 0 1 0 1 0 
0 0 1 0 1 1 0 0 1 0 0 1 0 1 

   ((a i. t) {"1 PH) # t 
s      
ff     

 
 
 
 
 
 
 
 
 
 
Chapter 8 Classification And Sets  77 

iiooa  
sngflf 

Incomplete classifications are also useful. For example, the classification provided by PH 
is  incomplete  because  the  space  belongs  to  none  of  the  classes.  Indeed,  every  n-way 
classification B implicitly defines a further class (which might be called other) defined by 
the expression -.+./B; that is, not the or over the classes. Any classification table may 
therefore be completed by applying the verb comp=: ] , -.@(+./) . 

Related classifications can be obtained from a table. Thus: 

   ]M=:>1 0 0 1 0;0 1 1 0 0 
1 0 0 1 0 
0 1 1 0 0 
   M *."0 1 PH 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 

   sovfop=: +./"2 M *."0 1 PH 
   sovfop 
1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 
0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 

   ((a i. t) {"1 sovfop) # t 
isiooa 
ff     

The  first    row  of  the  resulting  classification  table  sovfop  includes  sibilants  or  vowels; 
the second includes fricatives or plosives. 

For any classification table B, a corresponding disjoint classification can be obtained by 
suppressing from each column any 1 except the first. This is achieved by the expression 
</\B. For example: 

   </\PH 
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 
1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 
0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 

  
 
 
 
 
 
 
 
78  Arithmetic 

The last class of the resulting table represents “all consonants that do not fall in the earlier 
classes”. 

B. Sets 

A set is a one-way classification, and is therefore defined by a proposition. For example: 

   GT10=: >&10 
   L=: 2 3 5 7 
   MEML=: +./@(L&(=/)) 
   GT10 2 3 5 7 11 13 17 
0 0 0 0 1 1 1 

VOW=: +./@('aeiouy'&(=/)) 

III=: (]=<.) *. >&8 *. <&75   

   VOW 'happy those early days' 
0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 1 1 0 

   MEML i.15 
0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 

    III 6 7 +/ 2%~i.10 
0 0 0 0 0 0 1 0 1 0 
0 0 0 0 1 0 1 0 1 0 

Thus, VOW defines “The set of all vowels”, MEML defines “The set of all members of the 
list L (a parameter that may be changed) ”, and III defines “The set of all integers in an 
interval”. 

The proposition that defines a set is often itself defined in terms of the list of elements 
that belong to the set, as was done directly in the proposition VOW, and indirectly in the 
proposition MEML.  

Although we often speak loosely of the set as the list itself (as in “The set 'aeiouy'”, or 
“The  set  L”),  it  is  important  to  remember  that  the  definition  of  the  set  is  the  entire 
proposition, that the ordering of the elements of the list therefore imposes no ordering on 
the  members  of  the  set,  and  that  the  repetition  of  elements  in  the  defining  list  does not 
affect the definition of the set. 

A set is completely determined by the proposition that defines it, and we will sometimes 
speak loosely of “the set P” rather than “the set defined by P”. The defining proposition is 
often compound, and these compound propositions are often given special names. Thus: 

   PI=: P1 *. P2      The intersection of P1 and P2 

   PU=: P1 +. P2      The union of P1 and P2 

   PD=: P1 >  P2      The difference of P1 and P2 

  PSD=: P1 ~: P2      The symmetric difference of P1 and P2 

Although a proposition defining a set may have an infinite domain (such as all numbers), 
it is also useful to consider propositions restricted to a finite list of arguments. We will 
denote such lists by names beginning with U (for universe of discourse). 

For example, some or all of the letters of the alphabet might be assigned to colours, as in 
Acquamarine, Blue, Cyan, Dun, ... Orange, Pink, Quercitron, Red, ... Yellow, and Zaffer. 
The universe is then defined by: 

 
 
 
 
 
   U=:'ABCDEFGHIJKLMNOPQRSTUVWXYZ' 

Chapter 8 Classification And Sets  79 

and  the  sets  of  primary  and  secondary  pigment  colours  might  be  defined  by  the 
propositions: 

   P=: +./@(1 17 24&(=/)@(U&i.)) 
   S=: +./@(6 14 21&(=/)@(U&i.)) 

For example: 

   (P U)#U 
BRY 

U#~S U 

GOV 

   cv=: P U 
   cv 
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 

   ]ml=: cv # U 
BRY 

The vectors cv and ml defined above are the characteristic vector and member list of the 
set  defined  by  the    proposition  P  on  the  universe  U.  The  set  P  could  alternatively  be 
defined in terms of them: 

   P1=: {&cv@(U&i.) 
   P2=: +./@(ml&(=/)) 
   U#~P1 U 
BRY 

BRY 

U#~P2 U 

The table B=: #: i. 2^# U (whose rows are the base-2 representations of successive 
integers) provides an exhaustive classification of the universe U, including the empty set 
(represented by a characteristic vector of zeros), and the complete set (represented by a 
characteristic vector of ones). For example: 

   ]EC=: #: i. 2^# U=: 2 3 5 
0 0 0 
0 0 1 
0 1 0 
0 1 1 
1 0 0 
1 0 1 
1 1 0 
1 1 1 

This exhaustive classification is very useful. For example, the sums and products over all 
subsets of U can be obtained as follows: 

   +/"1 U*EC 
0 5 3 8 2 7 5 10 

*/"1 U^EC 

1 5 3 15 2 10 6 30 

Moreover, since EC is exhaustive, any collection of subsets can be obtained by selecting 
rows from it. For example: 

  
 
 
 
 
 
 
 
 
 
 
 
80  Arithmetic 

   5 1 2{EC 

(2=+/"1 EC)#EC 

1 0 1 

0 0 1 
0 1 0 

0 1 1 

1 0 1 
1 1 0 

C. Nub Classification 

The nub of an argument contains all of its distinct items. Thus: 

   nub=: ~. text=: 'mississippi' 
   nub 
misp 

   ]i=:nub i. text 
0 1 2 2 1 2 2 1 3 3 1 

         i{nub 

mississippi 

A  classification  of  an  argument  in  terms  of  its  nub  will  be  called  a  nub  or  self  or  auto 
classification. For example: 

   nub =/ text 
1 0 0 0 0 0 0 0 0 0 0 
0 1 0 0 1 0 0 1 0 0 1 
0 0 1 1 0 1 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 1 0 
   +/"1 = text 
1 4 4 2 

= text 

1 0 0 0 0 0 0 0 0 0 0 
0 1 0 0 1 0 0 1 0 0 1 
0 0 1 1 0 1 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 1 0 

The table on the right shows the use of the nub-classification monad = ; the expression 
+/"1  =  text  gives  the  distribution  of  the  items  of  its  argument,  that  is,  a  frequency 
count of its distinct items. 

D. Interval Classification 

A  list  of  integers  L  may  be  classified  according  to  its  interval,  that  is,  the  list  of 
successive  integers  beginning  with  the  largest  element  of  L  and  continuing  through  the 
smallest. Thus: 

' *' {~ (INT L) =/ L 

   (INT=: >./ - i.@>:@(>./ - <./)) L=:8 3 0 _1 0 3 8 
8 7 6 5 4 3 2 1 0 _1 
   (INT L) =/ L 
1 0 0 0 0 0 1             *     * 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
0 1 0 0 0 1 0              *   * 
0 0 0 0 0 0 0 
0 0 0 0 0 0 0 
0 0 1 0 1 0 0               * * 
0 0 0 1 0 0 0                *    

If  the  list  L  is  the  result  of  some  function,  then  the  foregoing  classification  is  called  a 
graph of the function. For example, if: 

 
 
 
 
 
 
 
Chapter 8 Classification And Sets  81 

    PARABOLA=: -&2 * -&4 

then PARABOLA i. 7 yields the list L used above. The foregoing results can be collected 
to define a graphing function as follows: 

   GRAPH=: ] =/~ >./ - i.@>:@(>./ - <./) 

Moreover,  the  expression  +./\GRAPH  L  produces  a  barchart of  L. Conversely, (in the 
case of non-integer values of L) it may be better to define a barchart function directly by 
substituting the comparison <:/ for the =/ used in GRAPH: 

   BARCHART=: ] <:/~ >./ - i.@>:@(>./ - <./) 

A graph may then be provided by the expression </\ BARCHART L. Finally, it may be 
remarked  that  a  barchart  is  a  classification  of  its  argument,  and  that  the  phrase  </\ 
applied to it produces the corresponding disjoint classification used as a graph. 

E. Membership Classification 

The functions VOW and MEML of Section B provide examples of defining a classification 
according  to  membership  in  a  list,  using  an  or  over  equality,  as  in  MEML=: 
+./@(L&(=/)) . Membership in a list is important enough to be accorded a primitive, 
denoted  in  mathematics  by  the  Greek  letter  epsilon,  and  here  by  e.  .  For  example,  the 
function MEML could be defined by e.&L . 

Membership can be used to define a form of plotting that supplements the barcharts and 
graphs provided by the interval classification in Section D. If B is a boolean table, then 
B{' *' gives a plot of the points indicated by the ones in B: 

   B 
1 1 1 0 0 0 
1 0 1 0 0 0 
1 0 1 0 0 0 

1 1 1 0 0 0 

B{' *' 

*** 
* * 
* * 

*** 

Such a table can be specified by the coordinates of its ones; for example, the coordinates 
defining B are the columns of the table: 

   b=:0 1 2 0 2 0 2 0 1 2,:0 0 0 1 1 2 2 3 3 3 

Laminate (,:) forms a table from list arguments: 

   b 
0 1 2 0 2 0 2 0 1 2 
0 0 0 1 1 2 2 3 3 3 

If A is a table of all coordinates of B, then B itself can be specified in terms of the index 
list  b  by  using  membership  (e.)  in  the  expression  A  e.  boxcol  b,  where  boxcol 

  
 
 
 
 
 
 
 
 
 
 
 
 
82  Arithmetic 

boxes the columns of its argument. We first define a function to generate all indices of a 
table, using the catalogue function { that forms boxed lists by choosing an element from 
each of the boxes in its argument: 

   ]w=:'ABC';'abcd' 
+---+----+ 
|ABC|abcd| 
+---+----+ 

   {w 
+--+--+--+--+ 
|Aa|Ab|Ac|Ad| 
+--+--+--+--+ 
|Ba|Bb|Bc|Bd| 
+--+--+--+--+ 
|Ca|Cb|Cc|Cd| 
+--+--+--+--+ 

   (i.&.>"1)  4 6 
+-------+-----------+ 
|0 1 2 3|0 1 2 3 4 5| 
+-------+-----------+ 

   ALLIX=: {@(i.&.>"1) 
   ALLIX 4 6  
+---+---+---+---+---+---+ 
|0 0|0 1|0 2|0 3|0 4|0 5| 
+---+---+---+---+---+---+ 
|1 0|1 1|1 2|1 3|1 4|1 5| 
+---+---+---+---+---+---+ 
|2 0|2 1|2 2|2 3|2 4|2 5| 
+---+---+---+---+---+---+ 
|3 0|3 1|3 2|3 3|3 4|3 5| 
+---+---+---+---+---+---+ 

We now use ALLIX to form the lists of coordinates in the usual form; that is, with the x-
coordinate  first  and  increasing  from  left  to  right,  and  with  the  y-coordinate  increasing 
from bottom to top:  

   ALLCO=: |.&.>@:|.@:ALLIX@:>: 
   ALLCO 4 6 
+---+---+---+---+---+---+---+ 
|0 4|1 4|2 4|3 4|4 4|5 4|6 4| 
+---+---+---+---+---+---+---+ 
|0 3|1 3|2 3|3 3|4 3|5 3|6 3| 
+---+---+---+---+---+---+---+ 
|0 2|1 2|2 2|3 2|4 2|5 2|6 2| 
+---+---+---+---+---+---+---+ 
|0 1|1 1|2 1|3 1|4 1|5 1|6 1| 
+---+---+---+---+---+---+---+ 
|0 0|1 0|2 0|3 0|4 0|5 0|6 0| 
+---+---+---+---+---+---+---+ 

   plot=: {&' *'@(ALLCO@[ e. boxcol@]) 

     boxcol=: <"1@|: 

   4 6 plot b 

 
 
 
 
 
 
 
 
 
 
 
Chapter 8 Classification And Sets  83 

***     
* *     
* *     
*** 

A  function  equivalent  to  plot  can  also  be  defined  by  replacing  all  of  its  component 
functions by the expressions that define them: 

  PLOT=:{&' *'@(|.&.>@|.@({@(i.&.>"1))@>:@[e.<"1@|:@])     

If  f and  g are two functions, then a plot of the points with x-coordinate  f  k{a and y-
coordinate g k{a will be called a plot of f with g or, alternatively,  a plot of g versus f. 
Thus: 

   f=: *: 
   (f ,: g) a 
0 1 4 9 
0 2 4 6 

g=: +:        a=:0 1 2 3 

   7 10 PLOT (f ,: g) a 

         *  

    *       

 *          

*       

F. Summary of Notation 

The monads self-classification and catalogue (= and  {), and the dyads membership and 
laminate (e. and ,:) were introduced in Sections C and E. 

Exercises 

A1  Enter  b=:  ?5  7$2 to produce a random boolean table, and n=:(7#2) #. b to 
produce  the  base-2  values  of  its  rows.  Then  enter  (7#2)#:  n  and  compare  the 
result with b . 

 A2  The base -2 value of the rows of the phonetic classification table PH is given by:   

n=: 258 2097184 41945216 71569476 62648250 

Use this fact to enter the table PH and then experiment with its use. 

B1  Define two or three propositions, and experiment with their intersection, union, and 

differences. 

B2  Predict  and  enter  the  complete  classification  table  for  four  elements,  and  select 

from it the classification table for all subsets of two elements. 

C1  Experiment with nub-classification on various arguments, including the boxed list 

;:'A rose is a rose is a rose.' 

D1  Enter the verbs defined in Section D, and experiment with them. 

E1  Predict and verify the result of {'ht';'ao';'gtw' 

  
        
 
 
 
 
 
            
            
            
            
 
 
84  Arithmetic 

E2  Plot -&2*-&4 versus ] on i.7, and compare the result with the parabola in Section 

D. 

E3  Plot 2&^ versus ^&2  

 
 
85 

Chapter 
9 

Polynomials 

A. Introduction 

A  polynomial  is  a  weighted  sum  of  non-negative  integer  powers  of  its  argument.  For 
example: 

   x=:1 2 3 4 5 
   e=: 0 1 2 3 
   c=: 1 3 3 1 
   x^/e 
1 1  1   1 
1 2  4   8 
1 3  9  27 
1 4 16  64 
1 5 25 125 

   +/"1 c*x^/e 
8 27 64 125 216 

c*x^/e 
1  3  3   1 
1  6 12   8 
1  9 27  27 
1 12 48  64 
1 15 75 125 

The  final  result  is  the  value  of  a  polynomial  with  exponents  e  and  weights  (or 
coefficients) c applied to an argument list x. 

A zero coefficient effectively suppresses the effect of the corresponding exponent (e.g., 
+/"1 (0 0 1 2)*x^/0 1 2 3 is equivalent to +/"1 (1 2)*x^/2 3 ); it is therefore 
convenient to express a polynomial only in terms of its coefficients c, and to assume that 
the corresponding exponents are i.#c : 

   POL=: +/"1 @ ([ * ] ^/ i.@#@[) 
   c POL x 
8 27 64 125 216 

The discussion in Sections A-D will be limited to polynomials with integer coefficients, 
but  general  polynomials  admit  real  and  complex  numbers,  as  discussed  in  Section  F. 
Because  a  general  polynomial  admits  an  arbitrary  number  of  arbitrary  coefficients, 
polynomials can be designed to approximate almost any function of practical interest. 

  
 
 
 
 
 
86  Arithmetic 

Although  its  utility  rests  largely  on  its  potential  for  approximation,  the  polynomial  has 
other important characteristics that can be discussed in the restricted context of integers: 
the following four functions are themselves polynomials: 

1.  The sum or difference of polynomials. 

2.  The product of polynomials. 

3.  The derivative (or “rate of change”) of a polynomial. 

4.  The integral of (or “area under”) a polynomial. 

Although  the  coefficients  of  the  polynomials  for  cases  3  and  4  are  trivial  to  compute 
(}.c*i.#c and 0,c%>:i.#c), their treatment will be deferred to Section H. 

B. Sums and Products 

The cases of the sum and product may be illustrated as follows: 

d=: 1 2 1 

   x=: 0 1 2 3 4 5 
   c=: 1 3 3 1 
   c POL x 
1 8 27 64 125 216 

   d POL x 
1 4 9 16 25 36 

   (c POL x) + (d POL x) 
2 12 36 80 150 252 

   (c+d,0) POL x 
2 12 36 80 150 252 

   (c POL x) * (d POL x) 
1 32 243 1024 3125 7776 

   TIMES=: +//. @ (*/) 
   c TIMES d 
1 5 10 10 5 1 

   (c TIMES d) POL x 
1 32 243 1024 3125 7776 

It will be more illuminating to discuss the sum and product of polynomials in terms of a 
table of an arbitrary number of coefficients. For example: 

   ]TC=: >1 3 3 1 ; 1 2 1 ; 1 1 
1 3 3 1 
1 2 1 0 
1 1 0 0 

   +/TC 
3 6 4 1 
   (+/TC) POL x 

 
 
 
 
 
 
 
 
 
 
Chapter 9 Polynomials  87 

3 14 39 84 155 258 

   TIMES/TC 
1 6 15 20 15 6 1 0 0 0 

   (TIMES/TC) POL x 
1 64 729 4096 15625 46656 

   TC POL"1 x 
1 8 27 64 125 216 
1 4  9 16  25  36 
1 2  3  4   5   6 

*/TC POL"1 x 

1 64 729 4096 15625 46656 

It should be noted that the final zeros appended to coefficients in forming the table TC do 
not change their effects as coefficients. However, it may be convenient to trim redundant 
trailing zeros from a result such as TIMES/TC above. Thus: 

   trim=: +./\.@* # ] 
   trim TIMES/TC 
1 6 15 20 15 6 1 

(i.7)!6 

1 6 15 20 15 6 1 

C. Roots 

If a function f applied to an argument a yields 0, then a is said to be a zero or root of f. 
A function is sometimes defined in terms of its roots. For example: 

   PIR=: */@(-~/) 
   r=: 2 3 5 
   x=: 0 1 2 3 4 5 6 
   r PIR x 
_30 _8 0 0 _2 0 12 

   r&PIR x 
_30 _8 0 0 _2 0 12 

(x-2)*(x-3)*(x-5) 

_30 _8 0 0 _2 0 12 

The  monad  r&PIR  is  also  said  to  be  a  polynomial  (or  polynomial  in  terms  of  roots) 
because  it  can  be  shown  to  be  equivalent  to  a  polynomial  c&POL  for  appropriate 
coefficients  c.  This  is  best  demonstrated  by  defining  a  function  CFR  that  produces  the 
coefficients from the roots. Thus: 

   AS=: #:@i.@(2&^)@# 
   AS r 
0 0 0 
0 0 1 
0 1 0 
0 1 1 
1 0 0 
1 0 1 
1 1 0 
1 1 1 

   POAS=: */"1@(-^AS) 
   POAS r 
1 _5 _3 15 _2 10 6 _30 

Boolean table of all subsets of #r items. 

Product over all subsets of -r. 

  
 
 
 
 
 
 
 
 
 
88  Arithmetic 

   CLBN=: =@(+/"1@AS) 
   CLBN r 
1 0 0 0 0 0 0 0 
0 1 1 0 1 0 0 0 
0 0 0 1 0 1 1 0 
0 0 0 0 0 0 0 1 

Classification by number of 
elements in set. 

   CFR=: +/"1@|.@(CLBN*POAS)    
   CFR r 
_30 31 _10 1 

Coefficients from roots. 

   (CFR r) POL x 
_30 _8 0 0 _2 0 12 
   r PIR x 
_30 _8 0 0 _2 0 12 

D. Expansion 

If the polynomial d&POL is equivalent to c&POL x+1, then the coefficients d are said to 
be the expansion of the coefficients c. More formally, d is the expansion of c if d&POL 
and c&POL@>: are equivalent. For example: 

   x=: i. 6 
   ]d=: +/ c * !~/~i.#c 
10 15 10 2 

c=:3 1 4 2 

   d POL x 
10 37 96 199 358 585 
   c POL x+1 
10 37 96 199 358 585 

   EXP=: +/@(] * !~/~@i.@#) 
   EXP c 
10 15 10 2 

   EXP^:4 c 
199 129 28 2 

   (EXP^:4 c) POL x 
199 358 585 892 1291 1794 

   c POL x+4 
199 358 585 892 1291 1794 

The definition of the function EXP will be analyzed in exercises. 

Although  the  function  EXP  and  its  non-negative  powers  can  produce  expansions  for  c 
POL x+i for any non-negative integer i, it must be modified to handle the general case 
for fractional values of i such as 0.1. This matter will be addressed in Section F, after 
the introduction of real numbers. 

 
 
 
 
 
 
 
 
 
Chapter 9 Polynomials  89 

E. Graphs And Plots 

Graphs  and  barcharts  of  functions  with  non-integer  results  can  be  produced  by  the 
methods of Section 8 D.We first define a uniform grid of a specified number of intervals, 
and use it to classify the non-integer results. Thus: 

   space=:(>./ - <./)@] % [ 
   grid=: <./@] + space * i.@>:@[ 
   graph=: {&' *'@ (</\@|.@ (grid </ ] + -:@space)) 
   10 graph %: i. 40 

                                    **** 
                             *******     
                      *******            
                 *****                   
            *****                        
        ****                             
     ***                                 
   **                                    
 **                                      

*        

The plots of Section 8 E may be extended similarly: 

  GPLOT=: [ PLOT |.@([ classify"0 1 ]) 

  classify=: <:@(+/@(grid </ ] + -:@space)) 

  PLOT=:{&' *'@(|.&.>@|.@({@(i.&.>"1))@>:@[e.<"1@|:@]) 

  6 10 GPLOT (*:,:+:) i.5 

          * 

       *    

     *      
* *         

F. Real And Complex Numbers 

In order to discuss further uses of polynomials, it will be necessary to extend the domains 
of our primitives beyond the integers to which they have been restricted thus far. 

Just as the inverse of the successor led to results outside of the counting numbers, so do 
inverses  of  certain  functions  on  integers  lead  outside  the  domain  of  integers.  For 
example: 

   a=: 1 2 3 4 

   *&2 ^:_1 a 

0.5 1 1.5 2 

   %&2 a 
0.5 1 1.5 2 

Rational numbers 

  
 
                                         
 
            
            
            
 
 
90  Arithmetic 

   %&2 -a 
_0.5 _1 _1.5 _2 

   ^&2 ^:_1 a 

1 1.41421 1.73205 2 

   %: a 

1 1.41421 1.73205 2 

Irrational numbers 

   %: -a 

Imaginary numbers 

0j1 0j1.41421 0j1.73205 0j2 

   a+%:-a 

Complex numbers 

1j1 2j1.41421 3j1.73205 4j2 

The  rationals  include  the  integers  and,  together  with  the  irrationals,  they  comprise  the 
real numbers. The informal extension of primitives to the real domain is straightforward; 
they are extended so as to maintain the properties discussed in Chapter 2. The imaginary 
and complex numbers are treated similarly, but merit further discussion. 

Since the square of any real number is non-negative, the square root of _1 must be a new 
number outside the domain of reals. It will be denoted by 0j1. The product of 0j1 with 
any  real  number  shares  the  property  that  its  square  is  a  negative  number.  This  follows 
from the normal properties of multiplication: 

   b=: 1 2 3 4 5 
   b*0j1 
0j1 0j2 0j3 0j4 0j5 

   (b*0j1) * (b*0j1) 
_1 _4 _9 _16 _25 

   b*b * 0j1*0j1 
_1 _4 _9 _16 _25 

   (b*b) * (0j1 * 0j1)  
_1 _4 _9 _16 _25 

   (b*b) * _1 
_1 _4 _9 _16 _25 

If  a  and  b  and  c  and  d  are  real  numbers,  then  a+0j1*b  and  c+0j1*d  are  complex 
numbers. Moreover, their sum can be derived from the familiar properties of addition and 
multiplication: 

   a=: 1+b=: 1+c=: 1+d=: 1 
   a,b,c,d 
4 3 2 1 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 9 Polynomials  91 

   (a+0j1*b) + (c+0j1*d) 
6j4 

   (a+c) + 0j1*(c+d) 
6j3 

   (a+c) + 0j1*(b+d) 
6j4 

6+0j1*4 

6j4 

The product of complex numbers can be derived similarly: 

   (a+0j1*b) * (c+0j1*d) 
5j10 

   ((a*c)+(0j1*0j1*b*d)) + (0j1*((a*d)+(b*c)))  
5j10 

   ((a*c)+(_1*b*d)) + (0j1*((a*d)+(b*c))) 
5j10 

   ((a*c)-(b*d)) + (0j1*((a*d)+(b*c))) 
5j10 

These processes can be described succinctly by representing each complex number by a 
two-element list, and using the primitive j. defined as follows: 

     j. y is 0j1*y 
   x j. y is x+j.y 
   j. b 
0j3 

4j3 

a j. b 

j./a,b 

4j3 

The  “complex  plus”  and  “complex  times”  functions  on  two-element  lists  can  now  be 
defined as follows: 

   cplus=: + 
   ctimes=: -/@:* , +/@([ * |.@]) 
   m=: 3 4 
n=: 5 2 
   j./m 
3j4 

j./n 
5j2 

   ]sum=: m cplus n 
8 6 

]prod=: m ctimes n 

7 26 

   j./prod 
7j26 

(j./m)*(j./n) 

7j26 

Although  a  collection  of  complex  numbers  could  be  represented  by  the  rows  of  a  two-
column table, it is more convenient to adopt an atomic representation, obtained by boxing 
each list. Thus: 

   M=:<m 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
92  Arithmetic 

   N=:<n 
   M,N 
+---+---+ 
|3 4|5 2| 
+---+---+ 
   < (>M) ctimes (>N) 
+----+ 
|7 26| 
+----+ 
As illustrated above, the verb cplus can be applied to these representations only by first 
applying > (open), and the corresponding atomic representation is obtained by applying 
the inverse < (box). 

The whole can be achieved by the conjunction &. in which the verb u &. v first applies 
v,  applies  u  to  that,  and  finally  applies  v^:_1.  The  conjunction  &.  is  called  under, 
because u is applied “under” v in the sense that surgery is performed under anaesthetic, 
the patient being restored from its effects at the end of the operation: 

   M ctimes&.> N 

+----+ 
|7 26| 
+----+ 
   M,N,M 
+---+---+---+ 
|3 4|5 2|3 4| 
+---+---+---+ 
   ctimes&.>/ M,N,M 
+-------+ 
|_83 106| 
+-------+ 

   CPLUS=: cplus&.> 
   CTIMES=: ctimes&.> 
   M CPLUS N CTIMES M 
+-----+ 
|10 30| 
+-----+ 
The monad magnitude (|) is extended to complex numbers to yield the square root of the 
sum of the squares of its imaginary parts: 

   | _5 
5 

   | 3j4 
5 

   %:+/*:3 4 
5 

In  other  words,  the  magnitude  is  the  distance  of  a  point  from  the  origin  when  the 
imaginary part is plotted against the real part. 

G. General Expansion 

 
 
 
 
 
 
The function EXP of Section D has the property that (EXP c) POL x is equivalent to c 
POL x+1. We will now define a more general expansion such that (y GEXP c) POL x 
is equivalent to c POL x+y: 

Chapter 9 Polynomials  93 

   x=: i. 6 
   y=: 0.1 
   c=: 3 1 4 2 
   GEXP=: +/@(] * !~/~@i.@#@] * [ ^ -/~@i.@#@]) 
   y GEXP c 
3.142 1.86 4.6 2 

   (y GEXP c) POL x 
3.142 11.602 41.262 104.122 212.182 377.442 
   c POL x+y 
3.142 11.602 41.262 104.122 212.182 377.442 

The definition of the expansion will be analyzed in exercises. 

H. Slopes And Derivatives 

If s is a small quantity, then the difference (f x+s)-(f x) gives an indication of the 
change in the result of the function f in the vicinity of the point  x. Moreover, the ratio 
s%~(f x+s)-(f x) obtained by dividing the “step size” s into this difference gives an 
indication of the rate at which f is changing. Because on a graph of the function this ratio 
is the slope of the secant line joining the points with coordinates x,f x and (x+s), f 
x+s, it is called the secant slope of f. For example: 

   f=: *: 

   x=: 4 [ s=: 2 
   (f x+s)-f x 
20 

The square function 

s%~(f x+s)-f x 

10 

   ]s=: 10^-i.5 
1 0.1 0.01 0.001 0.0001    

   s%~(f x+s)-f x 
9 8.1 8.01 8.001 8.0001  

We now define a dyadic function F such that s F x gives the secant slope of f at x with 
step size s: 

   F=: [ %~"0 1 f@([+/,@])-f@] 
   2 F x=: 4 5 6 7 
10 12 14 16 

   s F x 
     9      11      13      15 
   8.1    10.1    12.1    14.1 
  8.01   10.01   12.01   14.01 
 8.001  10.001  12.001  14.001 
8.0001 10.0001 12.0001 14.0001 

  
 
 
 
 
 
 
 
94  Arithmetic 

For a small step size, the secant slope s F x is a close approximation to the slope of the 
tangent to the graph of f at the point x, a value called the derivative of f at the point x. 
For example: 

Approximate derivative of square 

Approximate derivative of cube 

Approximate derivative of fourth power 

   s=:10^_10 
   s F x 
8 10 12 14 

   2*x 
8 10 12 14 
   f=:^&3 

   s F x 
48 75 108 147 

   3*x^2 
48 75 108 147 

   f=:^&4 
   s F x 
256 500 864 1372 

   4*x^3 
256 500 864 1372 

   n=:5 
   f=:^&n 

   s F x 
1280 3125 6480 12005 

   n*x^n-1 
1280 3125 6480 12005 

   n&([ * ] ^ <:@[) x 
1280 3125 6480 12005 

foregoing 

results  suggest 

The 
function 
n&([ * ] ^ <:@[). This relation will be explored by displaying the terms that must be 
summed to produce the results used in determining the slope, that is, f x+s and f x and 
(f x+s)-f x and s%~(f x+s)-f x. 

the  derivative  of  ^&n 

that 

the 

is 

For the power function f=:^&n and for the case n=: 3, the terms of f x+s are easily 
obtained from the direct expansion of the product (x+s)*(x+s)*(x+s) to the form : 

   ((s^3)*(x^0)+(3*(s^2)*(x^1))+(3*(s^1)*(x^2))+((s^0)*(x^3)) 

Thus for x=:2 and s=:0.1: 

  1 3 3 1 * (x^0 1 2 3) * (s^3 2 1 0)  
0.001 0.06 1.2 8 

Terms of ^&3 x+s 

  0 0 0 1 * (x^0 1 2 3) 
0 0 0 8 

Terms of ^&3 x 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 9 Polynomials  95 

  1 3 3 0 * (x^0 1 2 3) * (s^3 2 1 0)  
0.001 0.06 1.2 0 

Terms of difference 

  1 3 3   * (x^0 1 2  ) * (s^3 2 1  )               " 
0.001 0.06 1.2 
  1 3 3   * (x^0 1 2  ) * (s^2 1 0  )   
0.01 0.6 12 

Terms of slope 

  1 3 3   * (x^0 1 2  ) * (0^2 1 0  )   
0 0 12 

Slope for s=:0 

  1 3 3   * (x^0 1 2  ) *  0 0 1                        
0 0 12 

  3*x^2                                                                          
12 

 " 

 " 

In the general case of ^&n, the coefficients 1 3 3 1 and 0 0 0 1 become EXP CP n 
and CP n, and the difference becomes: 

   CP=: #&0,1: 
   EXP=: +/@(] * !~/~@i.@#) 
   CP 4 
0 0 0 0 1 

   EXP CP 4 
1 4 6 4 1 
   (EXP CP 4)-CP 4 
1 4 6 4 0 

   <@(EXP@CP - CP)"0 i. 6 
+-+---+-----+-------+---------+-------------+ 
|0|1 0|1 2 0|1 3 3 0|1 4 6 4 0|1 5 10 10 5 0| 
+-+---+-----+-------+---------+-------------+ 

   <@(_2&{.)@(EXP@CP - CP)"0 i. 7 
+---+---+---+---+---+---+---+ 
|0 0|1 0|2 0|3 0|4 0|5 0|6 0| 
+---+---+---+---+---+---+---+ 
It appears that the last two elements of the binomial coefficients of order n are n and 1. 
Since the binomial coefficients are the coefficients that represent the product (x+1)^n, 
insight can be gained by applying the product process of Section B to the corresponding 
coefficients 1 1: 

   1 1 */ 1 1 
1 1 
1 1 
   </.1 1 */ 1 1 
+-+---+-+ 
|1|1 1|1| 
+-+---+-+ 
   ]b2=:+//. 1 1 */ 1 1 
1 2 1 
   1 1 */ b2 
1 2 1 
1 2 1 

  
 
 
  
 
 
 
 
 
 
 
96  Arithmetic 

   </. 1 1 */ b2 
+-+---+---+-+ 
|1|2 1|1 2|1| 
+-+---+---+-+ 

   ]b3=:+//. 1 1 */ b2 
1 3 3 1 

I. Derivatives of Polynomials 

From the definition of the secant slope it is clear that the slope of a multiple of a function 
(m&*@f)  is  the  same  multiple  of  its  slope,  and  that  the  slope  of  the  function  f+g is the 
sum of the slopes of f and g. The same relations hold for derivatives.  

The  polynomial  c&POL  applied  to  an  argument  x  is  a  sum  of  terms  of  the  form 
(i{c)*(x^i) and (using the results of Section H) its derivative is (i{c)*i*(x^i-1). 
The  derivative  of  the  polynomial  c&POL  is  therefore  a  polynomial  with  coefficients 
}.c*i.#c. For example, using the functions F and POL of Sections H and A: 

   x=:1 2 3 4 5 
   D=: }.@(] * i.@#) 
   D c 
1 8 6 

c=:3 1 4 2 

(D c) POL x 
15 41 79 129 191 

   f=:c&POL 
   (s=: 10^-10) F x 
15 41 79 129 191 

J. The Exponential Family 

We  will  now  examine  coefficients  of  the  form  %!i.n,  and  their  relation  to  the 
coefficients of the corresponding derivative polynomial: 

   ]ce=: %!i.n=: 7 
1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 

   D ce 
1 1 0.5 0.166667 0.0416667 0.00833333 

Except 
(D ce)&POL agree, and the agreement improves as n increases. 

final  coefficient, 

function  ce&POL  and 

the 

the 

for 

its  derivative 

The primitive monad ^ (called exponential) is the limiting value of this polynomial. It is 
therefore  a  “growth”  function,  whose  rate  of  growth  is  equal  to  the  function  itself.  For 
example: 

   f=: ^ 
   f x 
2.71828 7.38906 20.0855 54.5982 148.413 

   s F x 

 
 
 
 
 
 
 
 
 
 
Chapter 9 Polynomials  97 

2.71828 7.38906 20.0855 54.5982 148.413 

Not only is the exponential important in its own right, but the odd and even parts of ^ and 
^@j. produce the hyperbolic functions (sinh and cosh, denoted by 5&o. and 6&o.) and 
the circular or trigonometric functions (sine and cosine, denoted by 1&o. and 2&o.). 

A  function  f  is  said  to  be  symmetric  or  even  if  it  gives  the  same  result  for  positive  and 
negative arguments; that is, if f and f@- agree. In terms of its graph we may say that an 
even function is “reflected in the vertical axis”. A function f is skew-symmetric or odd if f 
equals -@f@- or, equivalently, if f equals f&.- . Its graph is reflected in the origin. 

The functions: 

   e=: -:@(f+f@-) 

   o=: -:@(f-f@-) 

are,  respectively,  even  and  odd  functions.  Moreover,  e+o  equals  f,  and  they  are  called 
the even and odd parts of f. 

The adverbs ..- and .:- yield the even and odd parts of their arguments. For example: 

   cosh=: ^ ..- 
   sinh=: ^ .:- 
   ]x=: 0.2*i.6 
0 0.2 0.4 0.6 0.8 1 

 space must precede .. 

   cosh x 
1 1.02007 1.08107 1.18547 1.33743 1.54308 

   cosh -x 
1 1.02007 1.08107 1.18547 1.33743 1.54308 

   sinh x 
0 0.201336 0.410752 0.636654 0.888106 1.1752 

   sinh -x 
0 _0.201336 _0.410752 _0.636654 _0.888106 _1.1752 

   5 o. x 
0 0.201336 0.410752 0.636654 0.888106 1.1752 

   (sinh+cosh) x 
1 1.2214 1.49182 1.82212 2.22554 2.71828 

   ^ x 
1 1.2214 1.49182 1.82212 2.22554 2.71828 

The function ^@j. and its odd and even parts yield further important functions. We first 
observe that the magnitude of any result of ^@j. is 1. Thus: 

   2 3 $ ^@j. x 
                1 0.980067j0.198669 0.921061j0.389418 
0.825336j0.564642 0.696707j0.717356 0.540302j0.841471 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
98  Arithmetic 

   |^@j. x 
1 1 1 1 1 1 

As remarked in Section F, this implies that a plot of the imaginary part against the real 
part of any result of ^@j. lies on a circle whose radius has a length of 1. Moreover, the 
even and odd parts of ^@j. are its real and imaginary parts, and therefore the plot of one 
of the following functions against the other forms a circle: 

   cos=: ^@j. .. - 

   sin=: j^:_1@ (^@j. .:-) 

   26 52 GPLOT (sin,:cos) 0.2*i.30 
                    *    *    *                       
               *                   *                  
           *                            *             

       *                                    *         

    *                                          *      

                                                  *   
 *                                                    
                                                   *  
*                                                     

                                                    * 
*                                                     

 *                                                    

   *                                             *    

     *                                        *       

         *                                *           
             *                                        
                  *                   *               
                       *    *    * 

Moreover, (cos,sin) 0 is 1 0, and the length along the circle from this base point to 
the  point  with  coordinates  (cos,sin)  x  is  x.  Since  the  monad  o.  multiplies  its 
argument by pi, the circumference of the circle with unit radius is o. 2, and the sin and 
cos applied to the points o.4%~i.9 yield interesting results. Thus: 

   o. 2 
6.28319 
   sin o. 2 
_8.67362e_19 

   clean=: **| 
   clean sin o. 2 
0 

   ]p=:4%~i.9 
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 

   clean (cos,:sin) o. p 
1 0.707107 0 _0.707107 _1 _0.707107  0  0.707107 1 
0 0.707107 1  0.707107  0 _0.707107 _1 _0.707107 0 

 
 
 
 
                                                      
                                                      
                                                      
                                                      
                                                      
                                                      
                                                      
                                                      
                                                      
 
 
 
 
 
Chapter 9 Polynomials  99 

The monad * used in the definition of clean above is called signum:  *x is 0 if x is near 
zero, 1 if it is greater than zero, and _1 if it is less than zero.  

K. Summary Of Notation 

The  notation  introduced  in  this  chapter  comprises  complex  numbers  (3j4)  and  the 
corresponding verb j. (as in 3 j. 4 and j. 4); three conjunctions under, odd and even 
(&. .: ..); and six monads:  sine, cosine, sinh, cosh, signum, and exponential, (1 2 5 
6&o. *  ^). 

L. On Language 

In  accord  with  the  comments  in  the  language  section  of  Chapter  1,  notation  has  been 
introduced  sparingly,  only  as  needed  in  the  topics  under  discussion.  As  a  consequence, 
many important language constructs have been ignored. This section presents a sampling 
of them, grouped according to contexts in which they commonly arise. 

Programming.  Computer  programming  concerns  the  definition  and  use  of  verbs  in  a 
language executable on a computer, and programming therefore runs through this entire 
text. Nevertheless, it might not be recognized as such by programmers familiar with other 
languages, primarily because it is tacit rather than explicit. 

A tacit definition is one in which no explicit mention is made of the arguments to which 
the defined verb might apply. For example: 

   IQ=: <.@% 
   317 IQ 10 
31 

   IQ 0.166 
6 

Integer quotient of arguments. 

Integer reciprocal of argument. 

An explicit definition begins with an entry that includes the phrase 3 : 0, and follows 
with sentences that use x. and y. to denote the arguments, uses a colon alone on a line to 
separate  the  definitions  of  the  monadic  and  dyadic  cases,  and  concludes  with  a  right 
parenthesis alone on a line. For example: 

   iq=: 3 : 0 
if. y. < 0  
  do. 0   else. %: y. 
end. 
: 
<. x. % y. 
) 

   iq 
\ 25 
5 

   iq _25 
0 

  
 
 
 
 
 
 
 
 
100  Arithmetic 

   317 iq 10 
31 

Tacit  definitions  facilitate  the  use  of  structured  programming,  in  which  complicated 
functions are defined in terms of a hierarchy of simpler functions, each of which is useful 
in its own right. The following example is from statistics: 

Standard deviation 
Variance 
Normalization 
Mean 

std a 
0.816497 

mean a 

4 

   std=: sqrt@var 
     var=: mean@sqr@norm 
       norm=: ] - mean 
         mean=: +/ % # 
           sqrt=: %: 
   sqr=: *: 
a=:3 4 5 

   ]report=: ?3 4 5 $ 10 
1 7 4 5 2 
0 6 6 9 3 
5 8 0 0 5 
6 0 3 0 4 

6 5 9 8 5 
0 6 4 7 9 
7 2 0 7 3 
6 7 9 3 2 

9 7 7 6 0 
6 8 2 4 7 
4 2 2 3 1 
4 8 9 0 9 

   mean report 
5.33333 6.33333  6.66667 6.33333 2.33333 
      2 6.66667        4 6.66667 6.33333 
5.33333       4 0.666667 3.33333       3 
5.33333       5        7       1       5 

Mean over tables 

   mean"1 report 
3.8 4.8 3.6 2.6 
6.6 5.2 3.8 5.4 
5.8 5.4 2.4   6 

Mean over rows 

   std"1 report 
2.13542 3.05941 3.13688 2.33238 
1.62481 3.05941 2.78568 2.57682 
3.05941 2.15407  1.0198 3.52136 

Adverbs And Conjunctions. Adverbs and conjunctions may be defined either tacitly or 
explicitly. The following illustrates the tacit definition of adverbs: 

   ]a=: 1 2 3 4 5 
1 2 3 4 5 

   prsu=: \\. 

A sequence of adverbs  (prefix and suffix) 

   < prsu a 
+-+---+-----+-------+---------+ 
|1|1 2|1 2 3|1 2 3 4|1 2 3 4 5| 

 
 
 
 
 
 
 
 
 
 
 
 
Chapter 9 Polynomials  101 

+-+---+-----+-------+---------+ 
|2|2 3|2 3 4|2 3 4 5|         | 
+-+---+-----+-------+---------+ 
|3|3 4|3 4 5|       |         | 
+-+---+-----+-------+---------+ 
|4|4 5|     |       |         | 
+-+---+-----+-------+---------+ 
|5|   |     |       |         | 
+-+---+-----+-------+---------+ 

   +/ prsu a 
1 3  6 10 15 
2 5  9 14  0 
3 7 12  0  0 
4 9  0  0  0 
5 0  0  0  0 

   iprsu=: /\\. 
   * iprsu a 
1  2  6  24 120 
2  6 24 120   0 
3 12 60   0   0 
4 20  0   0   0 
5  0  0   0   0 

   inverse=: ^:_1 
   %: inverse a 
1 4 9 16 25 

q=: /prsu 
*q a 

1  2  6  24 120 
2  6 24 120   0 
3 12 60   0   0 
4 20  0   0   0 
5  0  0   0   0 

A conjunction with one argument 

   each=:&.> 
   <\a 
+-+---+-----+-------+---------+ 
|1|1 2|1 2 3|1 2 3 4|1 2 3 4 5| 
+-+---+-----+-------+---------+ 

   |. each <\a 
+-+---+-----+-------+---------+ 
|1|2 1|3 2 1|4 3 2 1|5 4 3 2 1| 
+-+---+-----+-------+---------+ 

   slope=: 1 : '[%~ + -&x.f. ]'  
   0.000001 ^ slope i.5 
1 2.71828 7.38906 20.0855 54.5982 

   ^ i.5 
1 2.71828 7.38906 20.0855 54.5982 

Explicit definition of adverb 

The  tacit  definition  of  conjunctions  will  be  illustrated  first  by  using  the  case  adverb-
conjunction-adverb, whose result can be used to provide the ordinary matrix product:  

   dot=: /@(("0 1)("1 _)) 
   m=:i.3 3 
   m 
0 1 2 
3 4 5 
6 7 8 

15 18  21 
42 54  66 
69 90 111 

m + dot * m 

  
 
 
 
 
 
 
 
 
102  Arithmetic 

A second illustration produces a conjunction that applies one of its arguments to a prefix, 
and the other to a suffix: 

   ps=: 2 : '(x.@{.)`,`(y.@}.)\' 
   f=: *: ps %: 
   3 f 2 3 4 5 6  
4 9 16 2.23607 2.44949  

f"0 1~i. 5 

0 1 1.41421 1.73205 2 

   1 f 2 3 4 5 6  
4 1.73205 2 2.23607 2.44949 

0 1 1.41421 1.73205 2 
0 1 1.41421 1.73205 2 

   f 2 3 4 5 6  
4 1.73205 2 2.23607 2.44949  

0 1       4 1.73205 2 
0 1       4       9 2 

Gerunds. The conjunction ` “ties” verbs together to form a gerund, a noun that (like the 
English  word  cooking)  carries  the  force  of  a  verb.  Gerunds  have  a  variety  of  uses,  of 
which two are illustrated below: 

   +`*/ 1 2 3 4 5 
47 
   1+2*3+4*5 
47 

   fac_or_sqr=: !`*: @. (>&5) 
   fac_or_sqr 8 
64 
   fac_or_sqr 5 
120 

   fac_or_sqr"0 i. 10   
1 1 2 6 24 120 36 49 64 81 

Insertion of successive verbs 

The conjunction @.(agenda) 
uses the index produced by  
its right argument to select a 
member of the gerund to  
produce the final result. 

Recursion. A function that is defined in terms of itself is said to be recursively defined. 
For example: 

   fac=: 1:`(] * fac@<:)@.* 
   fac 5 
120 

fac"0 i.6 
1 1 2 6 24 120 

The  1:  is  the  constant  function  that  yields  1,  and  the  monad  *  (signum)  yields  1  if  its 
argument is greater than 0. 

Controlled Iteration. If f and g are functions and h=: f ^: g, then x h y “iterates” 
f  by  applying  it  repeatedly  as  long  as  the  result  of  g  is  non-zero.  For  example,  an 
iterative  determination  of  the  square  root  using  Newton’s  method  may  be  defined  as 
follows: 

   h=: (-:@(] + %))^:([ ~: *:@]) ^: _ 
   5 h 1 
2.23607 

   *: 5 h 1 

 
 
 
 
 
 
 
 
 
 
 
 
 
5 

   1 2 3 4 5 h"0 (1) 
1 1.41421 1.73205 2 2.23607 

Chapter 9 Polynomials  103 

Linear Functions. The expression mp=:+/ . * uses the dot conjunction to produce the 
dot, inner, or matrix product mp. For example: 

   mp=: +/ . * 
   v=: i.3 
   m   
0 1 2 
3 4 5 
6 7 8 

m=: i. 3 3 
m mp m 
15 18  21 
42 54  66 
69 90 111 

   m mp v 
5 14 23 

v mp m 

15 18 21 

Moreover,  m&mp  is  a  linear  function  which  (as  stated  in  Section  2  D)  distributes  over 
addition. For example: 

   LF=: m&mp 
   a=: 2 3 4 
   LF (a+b) 
14 62 110 

   LF (m+2*m) 
 45  54  63 
126 162 198 
207 270 333 

b=: 5 1 1 
(LF a)+(LF b) 

14 62 110 

(LF m)+(LF 2*m) 

45  54  63 
126 162 198 
207 270 333 

Any linear function LF can be represented in the form M&mp for a suitable matrix M. If LF 
applies to vectors of n elements, then M may be obtained by applying LF to the identity 
matrix =i.n. For example, if p is an arbitrary permutation vector, then the permutation 
function p&{ is linear and: 

   n=: 6 

   LF=: p&{ 
   x=:2 3 5 7 11 13 
   LF x 
13 5 3 7 2 11 

   M=: LF =i.n 
   M&mp x 
13 5 3 7 2 11 

   M 
0 0 0 0 0 1 
0 0 1 0 0 0 
0 1 0 0 0 0 

]p=: n?n 
5 2 1 3 0 4 

%. M 
0 0 0 0 1 0 
0 0 1 0 0 0 
0 1 0 0 0 0 

  
 
 
 
 
 
 
 
 
 
   
 
 
104  Arithmetic 

0 0 0 1 0 0 
1 0 0 0 0 0 
0 0 0 0 1 0 

0 0 0 1 0 0 
0 0 0 0 0 1 
1 0 0 0 0 0 

   (%.M) mp 13 5 3 7 2 11 
2 3 5 7 11 13 

   M&mp^:_1 (13 5 3 7 2 11) 
2 3 5 7 11 13 

Exercises 

A1  Experiment with the expression  c  POL  x using  x=:i.7 and various coefficients 

c, including those from the columns of Pascal’s triangle in Section 7 C. 

A2  Using the value of x from Ex A1, evaluate (x+1)^n for various values of n, and 

compare the results with those of Exercise A1. 

A3  Define a function CP such that (CP n) POL x equals x^n. 

Answer: 

CP=: #&0,1:   

B1  Evaluate 1 1&TIMES ^:n  1 for various values of n. 

B2  Explore the definition of TIMES by evaluating the following: 

c=: 3 1 4 

  d=: 2 0 3 5  

c */d   

  </.c */ d      +//. c */ d 

Also compare TIMES with multiplication of integers in Section 4 C. 

B3  Use  theorems  3-5  of  Section  5  D  to  prove  that  the  product  of  polynomials  with 

coefficients C and D is equivalent to the polynomial with coefficients +//.C*/D. 

C1  Predict  and  test  the  results  of  CFR  n#1  for  various  values  of  n.  Repeat  for  CFR 

n#_1. 

C2  Define a function F such that n F r gives the coefficients of a polynomial having 

n repeated roots r. Test it on expressions such as 

5 F 1     5 F _1     5&F"0 -i. 6     F&_1"0>:i.6 

Answer: 

F=: CFR@#   

D1  Predict and test the results of EXP&CP n for various values of n, where CP is from 

Ex A3. 

D2  Explore the definition of EXP by defining the functions: 

A=: +/"1 

B=: ] * C 

C=: !/~@i.@#@] 

and then evaluating expressions such as C d=:3 1 4 2 and B d and A B d. 

E1  Predict and test the results of the following expressions: 

CTIMES/a=: 1 2;3 4;5 6 

CTIMES/\a 

 
 
 
    
 
 
 
 
     
  
 
 
 
 
 
 
Chapter 9 Polynomials  105 

a CPLUS CTIMES/a 

G1  Experiment with GEXP for various arguments. 

G2  Explore the definition of GEXP by defining the subtraction table function ST=: -

~/~@i.@#@] and evaluating ST c=: 3 1 4 2. 

G3  Evaluate y^ST c for various values of y, including 0.   

G4  Explain the equivalence of the expressions  (x+y)^n and  (y  GEXP  CP  n)  POL 

x, where CP is from Exercise A3. 

H1  Extend the sequence that concluded Section H. 

L1  Test the assertion that the scan +/\ is linear. 

L2  Predict and test the results of the following expressions: 

c=: 3 1 4 2 6 

+/\c 

I=: =/~i.#c 

M=: +/\ I 

d=: M +/ . * c 

(%.M) +/ . * d 

(>:/~i.#c) +/ . * c 

L3  Look  through  earlier  chapters  for  other  linear  functions,  and  re-express  them  as 
inner  products.  In  particular,  identify  the  cases  that  can  employ  Pascal’s  triangle 
(!/~i.n) and Vandermonde’s matrix x^/i.#c. 

L4  Predict and test the results of applying the matrix inversion function %. to some of 
the  matrices  used  in  Exercises  L2  and  L3,  and  use  them  in  defining  linear 
functions. 

L5  Examine  the  matrices  M  and  %.M  of  Ex  L2,  and  note  that  the  former  produces 

“aggregation” or “integration”, and the latter produces “differencing”. 

L6  Review  the  discussion  of  combinations  in  Section  7  C,  and  enter  and  experiment 
with  the  following  structured  definition  of  a  function  for  generating  tables  of 
combinations: 

        comb=: basis`[email protected] 

      basis=:i.@(<:,[) 

      recur=: (count#start),.(index@count{comb&.<:) 

         count=:<:@[!<:@[+|.@start 

         start=:i.@-.@- 

         index=:;@:((i.-])&.>) 
      test=: *@[*.< 

[Try 3 comb 4] 

  
 
 
 
 
References 

107 

1.  American Heritage Dictionary of the English Language, Houghton-mifflin (Any 

edition that includes the appendix of Indo-European roots). 

2.  Klein, Felix, Elementary Mathematics from an Advanced Standpoint, Dover 

Publications. 

3.  Cajori, F., A History of Mathematical Notations, Open Court Publishing Company, 

LaSalle, Illinois. 

4.  Lakatos, Imre, Proofs and Refutations: the logic of mathematical discovery, 

Cambridge University Press. 

  
 
110088 

  Arithmetic 

 
Index   

110099

0, 7 

1, 7 

action word, 3 

INDEX 

BARCHART, 83 

barcharts, 91 

base-10, 36 

addition, 5, 6, 10, 11, 12, 19, 35, 38, 54, 63, 92, 

bases, 36, 41 

105 

Addition, 5, 11, 36 

adds, 5, 42, 51 

adverb, 6, 10, 12, 13, 18, 22, 25, 26, 63, 65, 103, 

104 

adverbs, 3, 13, 22, 31, 99, 103 

ADVERBS, 12, 25, 103 

AHD, 13 

alternating sum, 16 

Ambivalence, 17 

ambivalent, 13, 17 

American Heritage Dictionary, 2, 109 

and, 60, 62 

annotated display, 6 

are, 3 

argument, 4, 5, 6, 8, 9, 10, 11, 12, 18, 19, 23, 28, 
29, 35, 40, 42, 46, 47, 50, 64, 72, 75, 82, 83, 
84, 87, 89, 98, 100, 101, 103, 104, 105 

Arithmetic, 9 

Arrangements, 69 

arrays, 42, 43 

associativity, 23 

Associativity, 18 

atomic, 68 

atop, 17, 22 

auto classification, 82 

base-value, 36, 41 

binomial coefficients, 71, 97 

bond conjunction, 17 

bond to, 17 

Bonds, 17 

Boole, 60, 63 

Boolean Dyads, 63 

Boolean Monads, 64 

Boolean Primitives, 65 

Boolean table, 89 

booleans, 55 

Booleans, 60 

Box, 30 

by, 15 

carrying, 37 

Catenate, 12 

Characters, 29 

circle, 100 

circular, 99 

classification, 28, 77, 78, 79, 80, 81, 82, 83, 85, 

86 

Classification, 77 

classified, 27, 78, 82 

clean, 100 

coefficients, 49, 87 

  
2  Arithmetic 

combinations, 108 

de Morgan, 11 

COMBINATIONS, 70 

decimal, 26, 35, 36, 37, 44 

commutative, 18, 19, 22, 38 

derivative, 96 

commutativity, 53 

Commutativity, 18 

derivative polynomial, 98 

Derivatives, 95, 98 

complex numbers, 22, 87, 92, 93, 94, 101 

derived verbs, 62 

Complex Numbers, 91 

diagonal adverb, 26 

computer, 1, 13, 15, 16, 22, 23, 32, 50, 101 

diagonals, 38 

Computer programming, 101 

dialogue, 1, 50, 51 

conjecture, 50 

dictionary, 2 

conjunction, 4, 15, 17, 22, 43, 94, 103, 104, 105 

differencing, 107 

conjunctions, 3, 14, 22, 101, 103, 104 

Display, 20 

Conjunctions, 4, 11 

CONJUNCTIONS, 12, 103 

Consonants, 78 

constant function, 105 

convolutions, 26 

coordinates, 84 

copula, 3, 11 

COPULA, 12 

copulative conjunction, 4 

correlations, 26 

cosh, 99 

cosine, 99 

distribute over, 19 

distributes, 105 

Distributivity, 19 

division, 23, 49 

divisors, 48 

domain, 3, 22, 28, 29, 49, 59, 60, 62, 65, 66, 80, 

91, 92 

Domain, 59 

dot, 105 

doubling, 3 

drop, 21 

duplicates, 18 

Counterexamples, 51 

counting number, 1, 2, 3, 5, 11, 47 

counting numbers, 1, 2, 3, 11, 28, 35, 91 

Counting Numbers, 1 

cross, 18 

CYCLES, 72 

cyclic repetition, 8 

dyad, 17, 18, 19, 21, 22, 23, 24, 27, 29, 32, 36, 

41, 42, 44, 61, 62, 63, 68, 72 

dyadically, 13, 41 

each item, 6, 37 

elementary algebra, 48 

Elementary Mathematics, 109 

empty, 21, 22, 47, 50, 81 

English, 3, 29, 64, 104, 109 

 
Index  3 

etymology, 2 

guesses, 50 

even, 2, 3, 9, 15, 16, 47, 49, 77, 99, 100, 101 

higher-rank, 42 

executable, 13, 101 

hyperbolic functions, 99 

exhaustive classification, 81 

identities, 21, 22, 48, 52 

Expansion, 90, 94 

identity, 4, 20, 21, 22, 24, 47, 52, 53, 54, 56, 62, 

experiment, 1, 13, 42, 50, 51, 85, 86, 108 

Experimentation, 22 

EXPERIMENTATION, 42 

explicit, 101 

Explicit definition, 103 

explore, 13 

exponent, 35, 49, 87 

exponential, 17, 98, 99, 101 

Exponential Family, 98 

exponents, 87 

factorial, 10, 42, 74 

false, 7 

formal proof, 47, 53 

fractions, 2, 22, 59 

fractured, 2 

Fricatives, 78 

73, 105 

Identity Elements, 21 

Imaginary numbers, 92 

in, 2 

indexing, 27 

Indo-European root, 2 

induction hypothesis, 56 

infinite, 2, 80 

infinities, 62 

infinity, 11, 22, 40 

Infinity, 21 

informal proof, 47 

inner, 105 

Insertion, 9 

inserts, 10, 42 

integer, 2, 15, 27, 28, 29, 47, 48, 59, 65, 67, 83, 

87, 90, 91 

function, 3, 50, 60, 83, 84, 85, 87, 89, 90, 95, 96, 

98, 99, 100, 104, 105, 106, 107, 108 

integers, 2, 3, 6, 7, 11, 16, 22, 23, 26, 28, 42, 44, 
47, 48, 49, 74, 80, 81, 82, 88, 91, 92, 106 

Generators, 64 

gerund, 104 

Grade, 28 

GRAPH, 83 

Graphs, 91 

greater than, 6, 28, 47, 54, 101, 105 

Greater-Of, 7 

greatest common divisor, 62 

Integers, 2, 35 

integration, 107 

Interval Classification, 82 

intervals, 27, 28, 91 

inverse, 2, 3, 11, 27, 28, 29, 31, 42, 43, 72, 74, 

91, 94, 103 

inverses, 15, 20, 23, 91 

Inverses, 20 

Irrational numbers, 92 

  
4  Arithmetic 

is, 3 

it, 3 

ITERATION, 105 

Klein, 109 

Lakatos, 50, 51, 52, 109 

Lakatos’, 50 

Language, 13, 23, 32, 101 

least common multiple, 62 

less than, 6, 9, 28, 54, 101 

Less than, 12 

Lesser of, 12 

Lesser-Of, 7 

linear, 19, 23 

linear functions, 107 

LINEAR FUNCTIONS, 105 

List, 7 

literal characters, 29 

Logic, 59 

magnitude, 23, 42, 43, 94, 100 

mathematical discovery, 50 

mathematics, 3, 10, 13, 49, 50, 52, 83 

matrices, 107 

matrix product, 104, 105 

max, 62 

maximum, 15 

Mean, 102 

MEMBERSHIP CLASSIFICATION, 83 

min, 62 

monad, 17, 18, 19, 23, 28, 30, 31, 42, 44, 47, 61, 
63, 64, 66, 70, 72, 75, 82, 89, 94, 98, 100, 
101, 105 

monads, 17, 21, 23, 25, 27, 31, 64, 65, 85, 101 

multiplication, 10, 11, 12, 16, 28, 35, 37, 38, 39, 

44, 47, 49, 53, 54, 92, 106 

Multiplication, 10, 37 

NAND, 66 

negation, 13, 65 

negative infinity, 22 

negative numbers, 2, 3, 11 

NOR, 66 

normal form, 37 

Normalization, 37, 39, 102 

notation, 1, 5, 12, 13, 22, 31, 42, 50, 54, 65, 74, 

101 

Nouns, 3 

nub, 82 

NUB CLASSIFICATION, 82 

odd, 15, 16, 45, 99, 100, 101 

Open, 30 

operator, 3 

or, 62 

over, 15 

pads, 31 

parentheses, 9, 40, 64 

Parentheses, 12 

partition, 31 

Partitions, 21, 25 

parts of speech, 3 

minimum, 7, 12, 15, 19, 22 

Pascal’s triangle, 71, 106, 107 

Mixed Bases, 41 

modulo, 29, 59 

Peano, 1, 2, 5 

 
permutation, 23, 27, 28, 67, 68, 69, 70, 71, 72, 

proposition, 60, 80, 81 

Index  5 

73, 74, 75, 105 

permutation vector, 27, 67 

permutations, 42 

Permutations, 67 

permuted, 19 

permutes, 47 

planes, 42 

Plosives, 78 

Plots, 91 

polyhedra, 51 

polynomial, 49, 87, 106 

polynomials, 26, 54, 87, 88, 91, 106 

Polynomials, 87, 98 

power, 4, 11, 12, 15, 22, 35, 39, 72, 75, 96 

Power, 11 

power conjunction, 4, 15 

predecessor, 2, 3, 5, 11, 13, 28 

Predecessor, 12 

prefix, 25, 104 

prime numbers, 16, 47 

primes, 26 

primitives, 62 

Primitives, 62 

product, 10, 38, 44, 47, 48, 53, 54, 56, 59, 88, 92, 

93, 96, 97, 104, 105, 106 

Products, 88 

propositions, 60 

Propositions, 60 

proverb, 4, 11, 20 

Proverbs, 3, 20 

punctuation, 9, 12, 64 

Punctuation, 9 

PUNCTUATION, 9 

quotes, 29, 31 

radices, 36 

range, 10 

Range, 59 

rank conjunction, 43 

rate, 95 

rational numbers, 59 

Rational numbers, 91 

ravel, 63, 65 

Real, 91 

recursively, 104 

Reduced Representation, 74 

redundant, 9, 70, 89 

re-entry, 13 

Refutations, 50 

Relations, 6 

remainder, 39 

remainders, 48 

programming language, 13 

repeated addition, 10, 12 

Pronouns, 3 

proofs, 47, 49, 50, 52 

Proofs, 45, 50, 52 

Properties Of Verbs, 17 

replicates, 8 

replication, 12 

representation, 36 

Representation, 35 

  
6  Arithmetic 

residue, 29, 31, 39, 40, 41 

successor, 1, 2, 3, 4, 5, 11, 28, 91 

RESIDUE, 28 

residues, 48 

right to left, 9 

Roman numerals, 35 

Roots, 89 

rows, 42 

Running maxima, 25 

Running products, 25 

secant line, 95 

secant slope, 95, 96, 98 

selection, 26, 27, 69 

Selection, 26 

Selections, 25 

Sets, 77, 80 

Shape, 12 

Sibilants, 78 

signum, 61, 101 

sine, 99 

sinh, 99 

skew-symmetric, 99 

Slopes, 95 

Sort, 28 

spread, 10 

square root, 49, 94 

Standard deviation, 102 

structured programming, 102 

Subtotals, 25 

subtraction, 5, 6, 11, 12, 13, 19, 64, 107 

Subtraction, 5 

subtracts, 5, 19, 23 

suffix, 104 

suffixes, 25 

Summary, 11, 31, 43, 65, 74, 85, 101 

SUMMARY, 22 

Sums, 88 

superscript, 11 

symbolic logic, 60 

symmetric, 19, 47, 99 

symmetry, 23 

Symmetry, 19 

synonym, 3 

Table, 7 

tables, 6, 7, 12, 15, 26, 38, 42, 43, 50, 52, 59, 63, 

65, 67, 68, 72, 102, 108 

tacit, 101 

tag, 2 

take, 21 

Tetrahedron, 51 

the counting numbers, 1, 3, 11, 91 

three-dot notation, 54 

train, 40 

trains, 40 

transposed, 63, 71 

trigonometric, 99 

true, 7 

truth-function, 60 

unbounded, 2 

under, 94 

universe of discourse, 80 

upon, 3, 11, 21, 77 

 
Index  7 

valence, 17 

Valence, 17 

Vandermonde’s matrix, 107 

variable, 3 

Verbs, 3, 17, 26 

VERBS, 12 

versus, 85 

Vowels, 78 

vectors, 52, 54, 72, 75, 81, 105 

word-formation, 30, 31 

verb tables, 7 

Verb Tables, 5 

verbs, 6, 10, 11, 12, 13, 15, 17, 18, 21, 22, 23, 25, 
26, 35, 39, 40, 41, 43, 59, 62, 63, 64, 65, 66, 
74, 86, 101, 104 

zero, 2, 3, 4, 7, 11, 23, 37, 40, 48, 49, 87, 89, 

101, 105