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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