Books/MathForTheLayman/Numbers
1A. The Counting (Natural) Numbers S1A.
The Counting numbers begin with 1 2 3 4 and go on forever, because every counting number has a successor that comes next after it. Thus:
>: 1
2
>: 2
3
>: 3
4
>: 4
5
>: >: 1
3
>: >: >: >: 2
6
>: 1 2 3 4 5 6 2 3 4 5 6 7
>: "performs an action" upon the number to which it is applied, and is therefore analogous to an "action word" or verb in English. In math, such a verb is also called a function (from Latin fungi, to perform or execute).
The number to which a verb applies may be called a noun. In math it is also called an argument, in the sense of a topic or subject ("You and love are still my argument" -- Shakespeare).
1B. Lists and Names S1B.
A collection of numbers may be written as a list, in the form 2 3 5 7 11, and verbs may be applied to such lists. For example:
>: 2 3 5 7 11
3 4 6 8 12
>: >: >: 2 3 5 7 11
5 6 8 10 14
A name may be assigned to a number or list by using the copula =:, and the name may then be used to refer to its referent. For example:
primes=: 2 3 5 7 11
>: primes
3 4 6 8 12
b=: >: primes
>: >: b
5 6 8 10 14
The main copulas used in English are "is" and "are". They may be used in reading mathematical expressions aloud, as in "primes are 2 3 5 7 11" for primes=:2 3 5 7 11, and "first is one" for first=:1.
Names may also be assigned to verbs. There appears to be no English word that corresponds to >: except, perhaps, the command "next" used to summon a successor from a queue. Thus:
next=: >:
next 5
6
next next 5
7
1C. Iteration (Repetition) S1C.
The expression next ^: 3 produces a verb that is equivalent to applying the verb next for three times. Thus:
primes
2 3 5 7 11
next ^: 3 primes
5 6 8 10 14
next next next primes
5 6 8 10 14
for=: ^:
next for 3 primes
5 6 8 10 14
list=: 1 2 3 4
next for list primes
3 4 6 8 12
4 5 7 9 13
5 6 8 10 14
6 7 9 11 15
1D. Inverse S1D.
The verb previous=: <: "undoes" the work of the verb next=: >:, and is said to be its inverse. Thus:
previous=: <:
primes
2 3 5 7 11
previous primes
1 2 4 6 10
next previous primes
2 3 5 7 11
previous next primes
2 3 5 7 11
previous 3
2
previous 2
1
previous 1
0
previous 0
_1
previous _1
_2
Because 1 is the first counting number, the zero (0) and the negative numbers (_1 and _2) shown above are not counting numbers. But they are integers, so called because they are unbroken or intact (in=not, tag=touch), as distinguished from the fractions (broken or fractured) referred to in the preface.
Adopting the seemingly-innocent notion of a verb that undoes the effect of next has, rather surprisingly, led us out of the original domain of counting numbers, and forced the adoption of a broader class, the integers, that includes the zero and negative numbers.
In adopting further inverses we will again experience the same need to broaden our domain, to include rational numbers, and imaginary numbers. Moreover, the surprising effect of inverse processes is not confined to mathematics.
Consider the effect of reversing a movie projector to run a film backward. If the film shows a locomotive moving forward along a track, the result of reversal is unremarkable. But if it concerns the dropping of an egg on the pavement, or a dive from a diving-board, the result is a startling illustration of the important distinction between reversible and irreversible processes.
Finally, the inverse of a function can be obtained by using iteration (for) with the right argument _1. Thus:
next for _1 primes
1 2 4 6 10
>: ^: _1 primes
1 2 4 6 10
>: ^: _1
<:
next for _3 _2 _1 0 1 2 3 primes
_1 0 2 4 8
0 1 3 5 9
1 2 4 6 10
2 3 5 7 11
3 4 6 8 12
4 5 7 9 13
5 6 8 10 14
1E. Addition and Subtraction (+ and -) S1E.
The effect of the verb next for 3 is to add 3 to its argument, and next for 3 primes is equivalent to the addition primes + 3 (using the familiar Saint George's cross to denote the verb). Thus:
next for 3 primes
5 6 8 10 14
primes + 3
5 6 8 10 14
0 1 2 3 4 5 + 0 1 2 3 4 5
0 2 4 6 8 10
+ table 0 1 2 3 4 5
+-+------------+
| |0 1 2 3 4 5|
+-+------------+
|0|0 1 2 3 4 5|
|1|1 2 3 4 5 6|
|2|2 3 4 5 6 7|
|3|3 4 5 6 7 8|
|4|4 5 6 7 8 9|
|5|5 6 7 8 9 10|
+-+------------+
The last result is an addition table, which may be "read" as follows:
To find the result of 3 + 4, choose the result (7) found in the row headed by 3 and the column headed by 4.
The verb + table is only one example of a function table, and other functions may be used. For example:
previous for 3 primes
_1 0 2 4 8
primes - 3
_1 0 2 4 8
- table 0 1 2 3 4 5 +-+----------------+ | |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| +-+----------------+
Exercises
Read from the addition table the sums 2 + 5 and 5 + 2 and verify that they agree. Make similar comparisons of additions of numbers that are similarly interchanged or commuted.
Because of the agreements noted in Exercise 1, addition is said to be commutative. Use the subtraction table to find whether subtraction is commutative.
1F. Bonding (&) S1F.
The verb + & 3 is equivalent to "add 3", that is, to next for 3. Thus:
+ & 3 primes
5 6 8 10 14
primes + 3
5 6 8 10 14
with=: &
+ with 3 primes
5 6 8 10 14
next for 3 primes
5 6 8 10 14
- with 3 primes
_1 0 2 4 8
primes - 3
_1 0 2 4 8
- with 3 + with 3 primes
2 3 5 7 11
+ with 2 primes
4 5 7 9 13
+ with 2 for 0 1 2 3 4 primes
2 3 5 7 11
4 5 7 9 13
6 7 9 11 15
8 9 11 13 17
10 11 13 15 19
Although the referent of primes is the list 2 3 5 7 11, it would not be correct to substitute the referent for the name in the foregoing expression, because the resulting 0 1 2 3 4 2 3 5 7 11 would be treated as a single list argument to for. Thus:
+ with 2 for 0 1 2 3 4 2 3 5 7 11 +&2^:0 1 2 3 4 2 3 5 7 11
The lists may, however, be separated by parentheses:
+ with 2 for 0 1 2 3 4 (2 3 5 7 11)
2 3 5 7 11
4 5 7 9 13
6 7 9 11 15
8 9 11 13 17
10 11 13 15 19
1G. Multiplication or Times (*) S1G.
Three times four (3 * 4) is said to be the addition of four copies (“plies”) of three. Thus:
3 + 3 + 3 + 3
12
3 * 4
12
* table 0 1 2 3 4 5 6 7 8 9 10
+--+--------------------------------+
| |0 1 2 3 4 5 6 7 8 9 10|
+--+--------------------------------+
| 0|0 0 0 0 0 0 0 0 0 0 0|
| 1|0 1 2 3 4 5 6 7 8 9 10|
| 2|0 2 4 6 8 10 12 14 16 18 20|
| 3|0 3 6 9 12 15 18 21 24 27 30|
| 4|0 4 8 12 16 20 24 28 32 36 40|
| 5|0 5 10 15 20 25 30 35 40 45 50|
| 6|0 6 12 18 24 30 36 42 48 54 60|
| 7|0 7 14 21 28 35 42 49 56 63 70|
| 8|0 8 16 24 32 40 48 56 64 72 80|
| 9|0 9 18 27 36 45 54 63 72 81 90|
|10|0 10 20 30 40 50 60 70 80 90 100|
+--+--------------------------------+
Exercises
Study the multiplication table, and comment on its properties (such as commutativity).
The numbers in column 5 are multiples of 5, that is, they result from multiplying by 5. Verify that they progress by "counting by fives", and check for similar properties in other columns and rows.
Comment on any patterns you find in the rows, columns, or diagonals of other function tables.
Make the table * table 2 3 4 5 6 6 7 8 9 10 and make a list of the counting numbers beginning with 2 (and up to perhaps 19) that do not occur in it. These numbers are called primes.
1H. Power and Exponent (^) S1H.
Just as repeated application of addition is equivalent to another important function (multplication, or *), so repeated multiplication is equivalent to power, or ^ . Thus, 3 ^ 4 is equivalent to 3 * 3 * 3 * 3. The right argument (in this case 4) is often called the exponent, and the expression 3 ^ 4 is read as “three power four” or “three to the power four”. For example:
3 ^ 4
81
3*3*3*3
81
0 1 2 3 4 5 ^2
0 1 4 9 16 25
^ table 0 1 2 3 4 5 +-+-------------------+ | |0 1 2 3 4 5| +-+-------------------+ |0|1 0 0 0 0 0| |1|1 1 1 1 1 1| |2|1 2 4 8 16 32| |3|1 3 9 27 81 243| |4|1 4 16 64 256 1024| |5|1 5 25 125 625 3125| +-+-------------------+
1I. Monomials and Polynomials (p.) S1I.
An expression such as 5*4^3 is called a monomial (one name) with the coefficient 5, the argument 4, and the exponent 3; a sum of monomials is called a polynomial (many names). For example:
5 * 4 ^ 3 320
(5*4^3) + (_2*4^4) + (1*4^1) _288
A polynomial in which the exponents in the successive monomials are successive integers beginning with zero, is said to be in standard form, and may be expressed using the polynomial function p., with the list of coefficients as the left argument. For example:
x=: 4
(2*x^0) + (3*x^1) + (4*x^2)
78
2 3 4 p. x
78
Exercises
Evaluate the following expressions, and comment on the results:
a=: 0 1 2 3 4 5
1 2 1 p. a
(a+1) ^ 2
1 3 3 1 p. a
(a+1) ^ 3
c4=: 0 1 3 3 1 + 1 3 3 1 0
c4 p. a
(a+1) ^ 4
1J. Division (%) S1J.
Division (to be denoted by %) "undoes" the work of multiplication. For example:
a=: 0 1 2 3 4 5 6
b=: a * 2
b
0 2 4 6 8 10 12
b % 2
0 1 2 3 4 5 6
b % 3
0 0.666667 1.33333 2 2.66667 3.33333 4
Just as the inverse of addition introduced new numbers outside the domain of the counting numbers, so some of the results of this inverse function lie outside of the domain of integers. These non-integral results (such as 0.666667) are "decimal approximations to" a new class of numbers, called fractions or rationals.
Just as we introduced a way to represent negative numbers, so we introduce a representation for rationals: 2r3 for the fraction two-thirds, 4r3 for the fraction four-thirds, etc. Thus:
1r3+1r3
2r3
a=: 0 1r2 1r3 1r4 1r5 1r6
a+a
0 1 2r3 1r2 2r5 1r3
a-a
0 0 0 0 0 0
a * a
0 1r4 1r9 1r16 1r25 1r36
+ table a +---+------------------------------+ | | 0 1r2 1r3 1r4 1r5 1r6| +---+------------------------------+ | 0| 0 1r2 1r3 1r4 1r5 1r6| |1r2|1r2 1 5r6 3r4 7r10 2r3| |1r3|1r3 5r6 2r3 7r12 8r15 1r2| |1r4|1r4 3r4 7r12 1r2 9r20 5r12| |1r5|1r5 7r10 8r15 9r20 2r5 11r30| |1r6|1r6 2r3 1r2 5r12 11r30 1r3| +---+------------------------------+
- table a +---+--------------------------------+ | | 0 1r2 1r3 1r4 1r5 1r6| +---+--------------------------------+ | 0| 0 _1r2 _1r3 _1r4 _1r5 _1r6| |1r2|1r2 0 1r6 1r4 3r10 1r3| |1r3|1r3 _1r6 0 1r12 2r15 1r6| |1r4|1r4 _1r4 _1r12 0 1r20 1r12| |1r5|1r5 _3r10 _2r15 _1r20 0 1r30| |1r6|1r6 _1r3 _1r6 _1r12 _1r30 0| +---+--------------------------------+
* table a +---+--------------------------+ | |0 1r2 1r3 1r4 1r5 1r6| +---+--------------------------+ | 0|0 0 0 0 0 0| |1r2|0 1r4 1r6 1r8 1r10 1r12| |1r3|0 1r6 1r9 1r12 1r15 1r18| |1r4|0 1r8 1r12 1r16 1r20 1r24| |1r5|0 1r10 1r15 1r20 1r25 1r30| |1r6|0 1r12 1r18 1r24 1r30 1r36| +---+--------------------------+
% table a +---+---------------------+ | |0 1r2 1r3 1r4 1r5 1r6| +---+---------------------+ | 0|0 0 0 0 0 0| |1r2|_ 1 3r2 2 5r2 3| |1r3|_ 2r3 1 4r3 5r3 2| |1r4|_ 1r2 3r4 1 5r4 3r2| |1r5|_ 2r5 3r5 4r5 1 6r5| |1r6|_ 1r3 1r2 2r3 5r6 1| +---+---------------------+
Exercises
Study the foregoing tables, and comment on their properties. (See the Exercises in Section 1G.)
Comment on the _ that occurs in the first column of the division table (it denotes infinity).
Study the multiplication table, and try to formulate rules for the multiplication of rationals.
1K. Review and Supplement S1K.
Using notation provided by a programming language (that will allow us to experiment with our math on a computer), we have introduced:
The counting numbers 1 2 3 4 etc.
The use of lists and of names that are assigned referents by a copula.
The iteration operator for=: ^: that applies a function for a specified number of times.
The function addition.
Its inverse (subtraction).
The class of integers that includes zero, and the negative numbers _1 _2 _3 _4 etc. introduced by subtraction.
The multiplication that is given by iterated addition.
The division that is inverse to multiplication.
The fractions or rationals that are introduced by division.
The power function and the polynomials based upon it.
Some of the assigned names (such as the for assigned by the expression for=: ^:) are "utilities" that will be utilized throughout the text, and are collected for easy reference in Appendix 1.
The pace of this text is brisk, moving on quickly to further topics as soon as the essential foundations for them are established. For example, although the Polynomials of Chapter 4 could provide a rich subject in itself, we pass on after a brief three pages to the Power Series of Chaper 5, and the Slope and Derivative of Chapter 6.
This gives a quick exposure to significant, and sometimes surprising, consequences of otherwise dull foundations. On the other hand, the reader may eventually (or immediately) want further treatments of the successive foundations: these are provided in a separate part of the book called Supplement.
In a printed text, the need to move between a given section and the corresponding supplemental section in a different part of the book might prove onerous. However, in reading the text from a computer (through a Browser), this switching back and forth is made by a click of a mouse.
For example, click on the S1K that appears to the right of the heading for this section to read the further discussion in the supplemental section S1K, and click on its heading to return.
The Find facility can be used as an index to find the occurrences of words in the text; invoke it by pressing F while holding down the control key, or select it from the Search menu. 1L. Notes S1L. On page 75 Hogben says:
It may therefore be soothing for many to whom mathematical expressions evoke a malaise comparable to being seasick, if they can learn to think of mathematics less as an exploit in reasoning than as an exercise in translating an unfamiliar script like Braille or the Morse code.
In this chapter we shall therefore abandon the historical approach and deal mainly with two topics: for what sort of communication do we use this highly space-saving – now international – written language, and on what sort of signs do we rely. To emphasize that the aim of this chapter is to accustom the reader to approach mathematical rules as Exercises in economical translation, every rule in the sign language of mathematics will have an arithmetical illustration …
He continues with:
In contradistinction to common speech which deals largely with the quality of things, mathematics deals only with matters of size, order, and shape. … First let us consider what different sorts of signs go to the making of a mathematical statement. We may classify these as:
punctuation;
models;
labels (e.g. 5 or x) for enumeration, measurement, and position in a sequence;
signs for relations;
signs for operations.
Conventional mathematical notation uses three pairs of symbols for punctuation: ( ) and [ ] and { }. We will use only the pair ( ) for this purpose, and will use the others for operations: { for indexing (selection), {. and }. for take and drop of a first item, and {: and }: for take and drop of a last item.
Hogben offers the following suggestions for study:
Although care has been taken to see that all the logical, or, as we ought to say, the grammatical rules are put in a continuous sequence, you must not expect that you will necessarily follow every step in the argument the first time you read it. An eminent Scottish mathematician gave a very sound piece of advice for lack of which many people have been discouraged unnecessarily. “Every mathematical book that is worth anything”, said Chrystal, “must be read backwards and forwards …”
---------------------
…Always have a pen and paper, preferably squared paper, in hand …. when you read the text for serious study, and work out all the numerical examples as you read …. What you get out of the book depends on your co-operation in the business of learning.
To this we may add the advice to use the computer in the manner introduced in the next chapter, and illustrated throughout the entire text. In particular, do not hesitate to do any computer experiments that may occur to you – the worst that can happen is the appearance of an error message of some kind, after which you may continue without any special action.