The new Mathematics, 1960

Mathematics

I R V I N G A D L E f e '

Instructor in Mathematics, School of General Studies, ' [ C o l u m b i a U n i v e r s i t y ;

Teachers, students, and general readers: This

is the first book to explain—In simple,

uncom-plicated language—the working methods, ideas,

and fundamental concepts of the revolutionary

developments in modern mathematics.

The Wonderful World of Modern Mathematics

In this era of expanding science, modem mathematics is

Hke a jet plane streaking across the sky to an ad

venturous destination. But to the average bystander who watches the jet-stream, even its take-off point

is a mysterious, hidden secret. The purpose of this

book is to help that bystander penetrate the un

known and fascinating realm of numbers real, imaginary, rational, and complex.

The author maps the routes of advanced theories from their fundamental origins, step by step, and shows how our system has grown from the whole num

bers we use in counting to measurements in space

that must be denoted by arrows.

Anyone who has a smattering of high-school algebra and geometry can follow the do-it-yourself sections at the end of each chapter and in the supplement. Pencil in hand, he will navigate through old

structures into new, and experience the thrill of

discovery that awmts him in the field of modem m a t h e m a t i c s .

T H I S I S A R E P R I N T O F T H E H A R D C O V E R E D I T I O N O R I G I N A L LY P U B L I S H E D B Y T H E J O H N D AY C O M PA N Y

O K W T

By Irving Adler

Other Books o£ Special Interest

H o w L i f e B e g a n b y f r y i n g A d l e r ,

A readable account of what science has i^ovet

muïrat'd."""" (#"IS69-35{)'

Magic House of Numbers by Irving Adler

Mathematical curiosities, riddles,

teach the basic principles of arithmetic. ( # KsoO

The Stars: Steppingstones into Space by Irving Adler

A clear explanation of the mysteries

with photographs and line drawings. (#1^ New Handbook of the Heavens &y

h»rd, Dorothy A. Bennett and «'J® A guide to the stars and planets. (#MD1X^5UÇJ

T o O u r R e a d e r s

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T H E N E W

M AT H E M AT I C S

With diagrams ly

I

Published by the new American library

Contents

® '«9 II

Info'" '?P"'l"9ed'iI''ànv'f°''' thereof, must

'" f <ty Company

^°«TP«mTrr,o,M„.i960

I. Numbers for Counting

n. Number Systems Without "Numbers'*

m. New Numbers From Old

IV. Numbers for Measuring V. Filling Out the Line

VI. Spilling Into the Plane VII. The Rank and Fûe: Matrices

Vni. Arrows That Are Numbers Bibliography

Summary of Basic Definitions

Do-Itr-Yourself Supplement I n d e x 7 1 3 3 5 U 7 3 9 0 1 2 5 1 4 3 1 6 0 1 8 1 1 8 2 1 8 4 - m

Foreword

AS WE go about our daily business, we malce fre quent use of whole numbers, like 1, 2, 3, and 4, which arise

when we count collections of objects. We also use fractions, like i and which arise when measurements are made. These are obviously two different kinds of number, because

fractions can never arise as a result of counting alone. In

what sense then, do they both deserve to be called num

bers? When we say that the fraction 4/2 is equal to the

whole number 2, what does this mean? How can a number of one kind be equal to a number of a totally different kind? These are among the questions which modern mathe matics has explored, and for which it has found answers.

Here are others:

We sometimes have occasion to use a number like the

square-root of two. It seems like a very elusive number that

is reluctant to show its face. Either it hides shyly behind a

symbol like or it reveals itself only piecemeal as a deci mal, 1.4, 1.41, 1.414, in a process that never ends. Why

doesn't it behave itself and settle down like a decent

ordinary fraction with a definite numerator and denomi

nator that can determine its value once and for all?

In elementary algebra we were introduced to negative numbers, and taught such mysterious rules as that the

product of two negative numbers is a positive number.

Where does such a rule come from?

The electrical engineer uses the number in the equations that describe the behavior of an alternating

ing''ila2na™''aL^rth^ "im^inary," yet there is

noth-describp Tf ^P1 the electrical current it helps to

althouch it i<i Ti^fiiiathematicians assure us,

âoT is the meaning of this

para-in thrcourse^? tn! ^"^stions that we shall look para-into

numbers of 1 } v? i^ske a close look at the

fact that our nP' answers in the

been gro^-ina wVl^^^ system has not been static, but has

number has % ' 1 conception of what constitutes a

cov^r The this growth, we shaU

dis->™. s*.sz.r„tf •

a t a n t t y e ^ p i d t e I t i s c o n

-w o r k s i n v e s t i g a t i o n , a n d

revolntTnl • concepts that are the fruit of a century-old

now ide^ i^a^ne """''T'l Associated with the

niatic»f^-f •! ™nni'ninny that gives modern

mathe-ticiartlT ohnracteristic flavor. To the

matheraa-a brkhMl^nw expressed by the new words serve matheraa-as

bûi penetrates to the core of a problem anH

o u J t b o u t t e ' n l ^ ' ' ' 1 " n n i

-a nefrXVcourse inT" T^'^nm-atics. It is not

rehash of old ideas W fn i niathematics. It is not a

tionally preseS an mtroduction to new ideas,

tradi-raathematics courses on^be° if ^ specialist, in advanced

But, although thfw^o "u Sraduate level,

elementary. Anybodv wbfk ^''/anced, the presentation is

geometry will be abL tf utXr Î f'® fand

_{A typical text in advaf^^fi f ®njoy this book,}

g m advanced mathematics today bristles

with such terras as group, ring, field, homomcrphism, iso morphism, and homeomorphism. These unfamiliar looking

words make it seem as though mathematics has abandoned

its old subject matter, and is no longer concerned with the study of numbers and space. This, of course, is not true.

■ Numbers and space are still very much at the heart of

mathematics. The new ideas and terms have arisen in con nection with a more penetrating analysis of their properties.

Underlying the terms group, ring, and field, for exam ple, are the old, familiar, simple operations of addition, sub traction, multiplication, and division. The mathematician has discovered that these operations are not the exclusive property of numbers alone. So he studies them in their most

general form, in order to discover rules that will be valid

in any context in which the operations are performed.^

The outlook of the modern mathematician is indicated

in his frequent use of the word stem "morph, meaning form, as in the words homomcrphism, isomorphism, and homeomorphism. The mathematician sees the number sys tem as a complex of interrelated structures. He studies these structures separately, and in their relationships to each other. The exploration of these structures has revealed

tliat we have, not a number system, but number systems;

not algebra, but algebras; not geometry, but geometries; not space, but spaces. While the properties of numbers and space have been generalized, the subject matter of mathe

matics has been pluralized.

The central thread around which the book is organized is the expansion of the number system, from natural num

bers to integers to rational numbers to real numbers to complex numbers. Although this sequence of steps in the development of the number system parallels very roughly historical stages in the development of the concept of num ber, the organization of the book is not chronological or historical. It is a logical organization from the modern point

of view, showing how the various number systems are^ re lated to each other. The development outlined here might

be referred to as "operation bootstrap." The system of 9

T

I

natur^ numbers (the whole numbers used for counting) : has defects that limit its usefulness. The story presented < here shows how mathematics has lifted itself by its own , bootstraps, using the defective system of natural numbers !! to construct bigger and better number systems that elimi

n a t e t h e d e f e c t s . I

At each stage of the construction of the expanded num- 1

ber systems, we encounter some of the structures such as

groups, rings and fields that receive so much attention in

modem mathematics. These modem concepts are intro duced first by means of familiar examples in the number

systems, and then other less familiar examples are given, '

too. As he reads the sections devoted to these modern con

cepts, the reader will be aware of the fact that he is merely nibbling at the comer of a great rug that has a beautiful but

intricate design woven into it. If what he sees from the

comer arouses his curiosity about the main design, it is ( hoped that he will satisfy this curiosity by systematic '

study from some of the standard text-books. A bibliography i i s g i v e n a t t h e e n d o f t h e b o o k . |

To get the most value and enjoyment out of this book,

read it with pencil and paper in hand. Verify the steps of each argument, work through all examples given, and do '

other examples like them. A "Do It Yourself" section at

the end of each chapter gives you an opportunity to strengthen through use your understanding of the new ideas

you will acquire in the book.

THE NEW MATHEMATICS

C H A P T E R I

Mumhers for Counting

WE ARE thoroughly familiar with the faces of the people with whom we live. Yet we are rarely conscious of the details of their features. If, as we look at a familiar face, we do take particular notice of the details, such as a curve of the lip, or a line in the forehead, it seems as though we are seeing them for the first time. Then, seeing these features that we never notice, we suddenly have the feeling that we

are looking at the face of a stranger. We shall have a similar experience with the familiar numbers of everyday life. When

We use these numbers, we take advantage of certain prop erties that they have. However, we are so accustomed to these properties that we are hardly aware of them as we use them. We shall now take particular notice of these properties, and list them explicitly. Looking at the familiar features of ordinary numbers, we shall see the strange new

face of modern mathematics.

The first numbers we all leam to use are those we need

to answer the question, "How many?" They are the num bers 1,2,3,4,5, and so on. There is an endless chain of these numbers. We use them for counting, apd we perform cal

culations with them, such as addition and multiplication.

Let us take a close look at these simple acta. Counting

Suppose that on Tuesday evening you want to see how

many days are left till the end of the week. It is likely that you will take count in this way: You will call off the names of the days, Wednesday, Thursday, Friday, and Sat

urday, and, for each day that you name, you will turn down

one finger on your right hand. After completing the list of

days, you find that you have turned down all the fingers on

your right hand except the thumb. So you conclude that there are four days left till the end of the week. "We find hidden in this procedure three important mathematical

concepts: the idea of a mapping, the idea of a onc-fo-ons correspondence, and the idea of cardinal number.

A mapping is a matching operation between two sets of objects: to each member of one set a member of the other

set IS aligned as partner. The two sets in this case are the

se of days being counted, and the set of fingers on your

hand. You set up a mapping when you single out a finger to turn down for each day you count. The mapping might be Bummarized in the following table: ^ &

Wednesday little finger Thursday —> ring finger Friday —+ middle finger Saturday —> index finger

Ue airowheads indicate that the mapping has a direction.

ÏTO select a finger for e^h day you name. This is not the

® T o . s p e c i f y t h e d i

rection of the mapping, we say that it is a mapping of the

t f e fl L

W e

r e f e r

t o

m a p p i n g '

^

-^i^sram below. In this

S e

s e t

o fl n l

i n t o

This mapping differs from the other on»» . .a

respect. Jhe two names, Richard and

mapped into the same number. This is an example of a many-to-one mapping, in which a single object may be the image of more than one object. In the mapping of days into fingers, however, no two days were mapped into the same finger. This is an example of a one-to-one mapping, in which

each object is the image of at most one object.

In the mapping of the set of days into the fingers of the right hand, one of the fingers, the thumb, wasn't used at all. For this reason the mapping of the set of days into the set of fingers on the right hand is not reversible. If we try to reverse it, we find that there is no mapping of the thumb into a day. We do not consider it a mapping then, because a mapping should provide an image for each object in the set on which the mapping is performed. However, if we consider only the set of fingers turned dovm, then the mai>-ping is reversible. Then, while each day named has a sepa

rate finger as its image, in the reverse mapping, each fing^

has a separate day as its image. In this case we say that the

two sets are in one-to-one correspondence. Two sets are in

one-to-one correspondence when there is a reversible map ping that assigns each member of one set to one and only one

partner in the other. The diagram below, usmg

doumheaded arrows, shows the onto-one correspondence^

e-tween the set of days and the set of fingers turned down.

Wednesday <—> little finger Thursday *—> ring finger

Friday <—> middle finger Saturday <—> "idex finger

When two sets can be put into one-to-one correspondence

by some mapping, we say that they contain ^be same num her of objects, or have the same cardinal number. AU sew that have the same cardinal number can be put into ^

one correspondence with each other. They form a y

of sets associated with that cardinal number. Eac car

number has its own family of sets. For exainple, se s ^

siating of single objects only belong to the ^ associated with the number we call one. Sets of pa

hlrtt ÎT?,"?® associated with the

num-ûssocîafpH triples belong to the family of sets

Anv t? r ^ we call three, and so on.

When wp belongs to one of these families,

in the set?" if ' ^ 'How many objects are there

does t b oiftÔ'.-'S' of oets

procedurrw. • 1, question, we foUow this

^ s t ^ d

u s e

i t

O S

We are intprpcfprl ■ conapansons. We match the set

find one with whiS standard sets, until we

spondence In thîo tp can he put into one-to-one

corre-it belongs to and thp^P identify the family of sets that

family. This'is Drepî<!Pii?^ i"t associated with that

a g a i n s t fi n g e r s Y o u m a t c h d a y s

finger alonf i a toCdsP^f^

You use the set consS ofliuipT"''^'

standard set to represent fho ^ finger as a consisting of fittle finger rine

a standard set to represent tlfp ^ ù "^^ddle finger as

sisting of little finger rine finr

finger is your standaM set refrese^w tf

That IS why you drew the concS^ .the number four.

o l

e n n V o \ t t

» o r e s o p S e T r « > a t i s

out four objects by .ayiugnroSvS

lour. As we count, we set up a opp ®' ^^0, three

beU^en the objects we are cZtinl '^"''''spondenco

number-names. The first om!p!^ - "® sets of

consisting of t>ip e;« ^ ^l^ct is matchpd « 'ii. spokeu

matched wUh tte S® """o." Ue « f

The first three Lota r I ^ords

words, "one, two, thtL" An 7"'* ">

■

= ^ot eontr

names of the numbers in n^1 1° ?"• % UsinL fv,'"® °f fbe

Btandard set step by sten TOu° ®'^o, we keo number

that the last nuLbTr4tetLd

the last standard set against which we matched the objects we are counting. So it is also the cardinal number of the

counted objects. By using standard sets made up of number

names arranged in order we telescope a whole series of

matching operations into one, and end up with the answer

to the question, "How many?'*

A d d i t i o n

A typical problem in addition is to find the sum of the

numbers 2 and 3. The meaning of this problem can be re stated in terms of sets of objects in this way: Suppose you have one set of objects whose cardinal number is 2, and

another set of objects, different from those m the first set,

and whose cardinal number is 3. A larger set is formed when these two sets are united. What is the cardmal num

b e r o f t h e u n i t e d s e t ? , , , • v

We can answer this question by actually forming such a

united set, and then identifying the standard set wi^ which

it can be put into one-to-one correspondence, i his is the procedure of the beginner in arithmetic, who first turns

down two fingers, then turns down three more fingers, an

finally matches the set of turned down fingers against the

standard set consisting of the spoken words, one, wo,

three, four, five." However, experienced calculators use a short-cut for getting the answer. Having carried out the process of uniting and counting sets of objects many before, we record the results in an addition tab e w ic ^ memorize. Then, any time we want to find the sum o w numbers, we don't have to manipulate sets of objects again.

W e m e r e l y c o n s u l t t h e t a b l e . _

Using the addition table instead of adding on our fingers

is more than just a time-saving convenience. wt,rr» abstraction that/iQ-s c/ian(7ed the wi€amn<7 o/a . we add on our fingers, we are actually working wi

numbers, which are properties of sets of objects, \\hen we

use the addition table, we are performing an operation on

abstract symbols. This operation can now be performed

closely. abstract operation a little morei

call numbers wl 3, 4, and so on, that we

another one the fX ^"d ^en pict

bera anothe^ t^mbe ^P''- °f

pick the number 2 first "nd fh '""'J"' ™

assigns as their sum v ^^umber 3 next, the table

of numbers 2 and 3 in a dSn-[ Picked the pair

two is first and the 3 it: « j o specifying that the

ordered pair (2 3). Whafth n-®

a definite number called thp^ ^"^ition table does is assign

fu^bers. When wrLseribe

^e^ sTof oTd^faS

up more clearly if we write it thi^^way; ^ ^^PP"^e shows

T u

The plus sign printed over tho «

™ r ô : : f s a ™ f e f t h e

Pnmbers into peopkt nam ""S^d, wo coX m'' 'be

mappings, because we eL m''- bave a „;?''P,P^'m of

Other set of objects in ^uy set of u- ® ®boice of

We observe, first, that the numbers we are pairing off m

our ordered pairs are selected from the list of symbols, 1, 2, 3 4 etc. To distinguish this set of symbols from the cardinal numbers from which they were derived, let us give it a name. We shall call it the system of natural numbers. We observe next that the number assigned as sum to each

ordered pair is selected from the same set, the system of

natural numbers. A mapping which assigns to e^h ordered

pair of objects in a set another object selected from the

same set is called a binary operation. So addition is an ex

ample of a binary operation defined on the system of natural

n u m b e r s . . , , , . i i a — f t

If we consult the addition table, we find tha »

2 + 3 = 5, 3 + 2 = 5, and 4+1 = 5. The different ordered

pairs, (1, 4), (2, 3), (3, 2), and_(4, 1) are aU mapped mto

the same image, 5. So addition is a many-to-one mapp g.

In particular, an ordered pair like (2,3) and the C » J »

obtained by having the 2 and 3 change ^ . the same image. We could wnte 2 + 3 — 3 + . simi

statement is true for the sum of every ordered pair of uatur^

numbers. We find that 5 + 2 = 2 + 5, 9 +16 = 16+ ^

etc. This characteristic of the addition of natural numbera

can be summarized in the following rule: If e e

stands for any natural number, and the ,,

any natural number, then a-i-b = b-¥n. That is, ^

natural numbers being added commute or ^ ' the sura is still the same. So this rule is known ^ ®

-mutative law of addition, and we say that a i

natural numbers is a commutative operation.

We are so accustomed to using the comnm a ive

addition that it may seem to be obvious, and hardly worm

mentioning. But it needs special mention , some binary operations, like addition of '

obey a commutative law, there are others that ^

example, one of the operations we

school arithmetic is called division, and is en .

symbol -t-. It is not a commutative operation, because the

wiSoS^chanpfn^^th^^ cannot, in general, change places^

Without changing the result. For example, 8 - 2 is Lt equal.;

tion perform^ Tn about it so far, is an opera-i

it to three numb^rd numbers. We can also extendi adding two of thpm ' three numbers by first |

HowelerT?oftSee the sum to the third,

mte orde'r, 4 have in a

defi-^ght add the sum of 2 3 ^ + T '

the sum of 3 and 7 might add 2 to > down in this form' (9 4. t\ P°®®^hilities can be written

-notation, the narentw • j- 2 + (3 + 7). In this

first. When we can^ ^ be found

doesn't make any^ffereL *bat it

fuse the results^omro" t first,

be-I = 12, and 2 + (3 + 7i = oi (2 + 3) + 7 = 5 +

teristic of the addition of tht- ® "" ^2. This is a charac- |

what numbers are used It i ""'tural numbers, no matter

+ c = a + (6 + ™ed in the rule, (a + b)

umbers. This rule savs thôt '. ?, " stand for any natural ,

d't.on we are free tHl?• . S"-®' step of the ad- "i

With the number on the wî ^ number either

■

S So the rule is known t I ™»ber on the

Mt.?«t^ we say that addition of ® assoaative law 0/

addi-amative operation. sC n " . ""®bers is an

as-:+"b Vo=: ?

Cwhi?S?=^"-X -untion

be-yîlfi all'tr txîlT sle bttS""/ ^"''1

agnate the oner^H^ we useThp^ not

«Puration th?t e^nl" nll^ no to

des-bers and fracHn« *l P ^^o*"med on tho ( "ï- ^ binary

20 Pt 8 aSd le! P°'P«on,

Bymbol 12 av 12 means the average of 12 and 12, which is

also 12. The symbol 16 av 12 means 14, and the symbol

8 av 14 means 11. This operation does not obey an as

sociative law, because (8 au 16) au 12 is not equal to 8 au (16 au 12). In fact, (8 av 16) av 12 means 12 au 12, or 12, while 8 av (16 av 12) means 8 av 14, or 11.

By a step-by-step process, the use of the commutative

law and the associative law for addition of natural numbers •can be extended into a general rule for the sum of any

finite selection of natural numbers: When you add a finite

selection of natural numbers, you can list them m any

Order, and group them as you please. The sum will always come out the same.

Multiplication

The meaning of multiplication of natural numbers, like

the meaning of addition, can be stated first in terms of sets of objects. To multiply 2 times 3, we set up a rectangu lar array of objects, consisting of two rows, with three ob jects in each row. Then we find the cardinal number of

this set. In general, to multiply the numbers a and b, we

find the cardinal number of a set consisting of a rows with

b objects in each row. The answer is called the product o

a and b, and is designated by a • b, where we use a

the symbol for multiplication. Once we have fo^i^d the product of two natural numbers, we can record it for future reference in a multiplication table. Then we can separate the operation of multiplication from its original meaning of finding the cardinal number of a rectangular array^ of objects. We can think of it instead as merely a mapping of ordered pairs of natural numbers into the system of natu ral numbers. We usually show the mapping by a scries of

statements like this: 2 • 3 = 6, 2*4 = 8, 2 • 6 = 10, etc.

S X p o f " ™ ™ " - î ' J '

(2,3)—L^ 6

(2.4) (2.5)

e t c .

~ t S f a ' P -

_{a natural number ^vf ™^PP^"S is al-}

o f

binaiy operation on the systeî nf + ? addition, is a

know from our exneripnr! -T numbers. We

numbers that 2 • 3 = 3 | f^nltipUcation of natural

a . (b . c) = Ca M . . mu-' <^sociative

the fact that 2 - (3

■

5) = 2 • Ï5 - 30in

6; 5 = 30. Because of this law wr^n ""v ' 3) • 5 =

0 three numbers without parektheseraJiS '• P^^^uct

nite meaning; a - b • c^a - (k n Sive it a

defi-bining the commutative law ~ C® " 6) • c Com

crfe.td tup rts"r-,^yo:t Cth?

"TL^oTs Te t Product wTuT

1 0

write 9 = 5 + 4. So the number of stars in the rectangu lar array can also be written as 3 ■ (5 + 4). Now, suppose

we move the first five stars in each row over to the left, so

that a space separates them from the rest of the stars in the same row. The effect is to split our rectangle into two rectangles. One rectangle has three rows with five stars in

each row, so it contains 3 • 6 stars. The other rectangle has

three rows with four stars in each row, so it contains 3 • 4

stars. Since we get the original rectangle by uniting the two smaller rectangles, the number of stars in the original rectangle is the sum of the numbers of stars in the two smaller rectangles. This fact is expressed in the statement that 3 • (5 + 4) = (3 * 5) + (3 • 4). We can verify the

cor-* cor-* cor-* » » * » * *

« « n b ♦ »♦ ♦ » v n M w f t h *♦ • »

rectness of the statement by noting that 3 • 9 = 27, and 15 + 12 = 27. In general, if o, 6, and c, stand for natural numbers, a • (6 + c) = (o • b) + (a * c). Similarly, (b + c) • o = (b • o) + (c • a). This rule is known as the distrib-utive law and expresses the fact that multiplication is distrib utive with respect to addition. That is, the multiplier can be distributed among the individual terms in the expression it

multiplies. In the statement of this law, multiplication and

addition cannot change places. While 3 + (5 • 4) = 3 + 20 = 23, (3 + 5) • (3 + 4) = 8 • 7 = 56, so that addition is not distributive with respect to multiplication. It is cus tomary, in writing an expression like (a • b) + (a * c) to leave the parentheses out, so that it looks like this: a • b + a.' c. In such an expression, which gives instructions for doing both multiplication and addition of some numbers, it is understood that the multiplications must be done first.

The Five Laws

We originally introduced the natural numbers as symbols

for the cardinal numbers. Then we made these observa

tions about them: There are two binary operations defined

on the natural number system, and we call them addition

embodied in the addition and ir^r ^ operations are |

amining these tables we foun?"fi '

by the natural number system fh^ ^

sociative laws of arlHifinn +k^ commutative and as- I laws of multiplication nnH c commutative and associative

serts that multiplication 'J A' which as- '

addition. These laws havp n with respect to , velopment of our notion of significance in the de- ;

■

when we carry ourc««T ^^ ^ that

iiumbers. It is enough to th r, W *1 as cardinal

related to each other by additir? ^ abstract symbols ;

that obey these five laws Th* multiplication tables '

define numbers as follows* Anu^ suggests that we re- |

0 / o b j e c t s o n - w h i c h ^ i

and multiplication aZ opcrahbns called

^ commutative and assodative i- addition

commuta-^^^pect to addition. « distributive with

b» systei aS lo o„ In to 7"' ^ large' nT

systematic construction nf i of thi«i K ? .

terns, using the ^ and W ^ the

? a ? e O f i o a s y s - ^

IV. {a-l)'C = a • Q)-c)

V. c • (6 + c) = a • b + a ■ c

or (6 + c) • a = b • Û + c • a

Large and Small Numbers

The natural number system bas some otber important characteristics besides the five laws. One of these is that

we can compare any two numbers in it for size. The num^r 6 is larger than 4, and 4 in turn is larger than 3. The^ notion of larger and smaller is derived from addition in this way. We say that b is larger than a if 6 is equal to the sum of a

and some other natural number. For example, 5 is larger

than 4, because 5 = 4 + 1; 5 is larger than 3, because

5 = 3 + 2.

One System with Many Disguises

There are many different ways of writing the natural

numbers. In the system of Arabic numerals that we use

every day, the numbers one, two, three, four and five are written as 1, 2, 3, 4, 5. In Roman numerals, still used on clock faces and monuments, they are written as I, II, III, IV, V. In Hebrew they are written as the first five letters of the alphabet. If we think of these different systems of numerals as symbols for the cardinal numbers, then they

are different ways of representing one and the same num

ber system. However, we may also think of each system of numerals as a separate number system in its own right, with addition and multiplication defined by its addition and multiplication tables. The Arabic, the Roman, and the

Hebrew numerals could then be referred to legitimately as

three separate number systems. But they are number

^s-tems that can be used interchangeably, so, although they are separate systems, they are still somehow the same. In the next chapter we shall encounter number systems that are not interchangeable and may not be considered tho same. In order to recognize when number systems are inter changeable, and when they are not, we have to define what We mean when we say different systems are the same.

^at we have in mind is that they have the same strwcfwre.

•tor two number systems to have the same structure, each

number in one system must have a counterpart in the other system. We can express this requirement in technical language by saying that there must be a mapping of one

system into the other that places them in one-to-one cor

respondence. But the one-to-one correspondence alone is

not enough. We want to be sure, too, that the results of computations in one system correspond to the results

.of computations in the other system. So we say that two number systems have the same structure, or are isomorphic, (1) there is a mapping of one mto the other that puts them into one-to-one correspondence, and (2) under this snapping, sums and products are preserved. The require-I

ïuent can also be stated in this way: Under the mappimr

e^h element in one system "nage, in the other'

Moreover, the image of the sum of two riumbers is the sum

_{of the images; and the image of the product is the product}

of the images. Comparing Arabic numerals and Romnn

numerals, for example, we can set up a one-to-one en?

respondence, shown in part in this table:

Each ^stem has its own addition and multinli,. +•

^ f is shown in the customary Ln "

ïnents below: ^ ®^^are arrange^

Addition 2 3 + 1 2 3 1 2 3 4 2 3 4 5 3 4 5 6 + _I n _m I _{I I} I I I _{I V} I I _{I I I} I V _V I I I _{I V} V _{V I} MuUiplication • 1 2 3 • I I I I I I 1 1 2 3 I I I I I I I 2 2 4 6 I I I I I V V I 3 6 I I I I I I V I I X

Under the mapping the image of 2 is 11, and the image of 3

is III. The sum of 2 and 3 is 5. The sum of II and III is

V, which is the image of 5. So the sum of the images is the image of the sum. The product of 2 and 3 is 6. The product of II and III is VI, which is the image of 6. So the prod

uct of the images is the image of the product. Arabic nu merals and Roman numerals, considered as separate num

ber systems, are isomorphic to each other. Although numbers in one system look different frqm numbers in the

other system, the relationships within Ihe systems, as ex

pressed in the addition and multiplication tables, have the

same structure. So the two systems are really only one

structure appearing in two different styles of dress.

Z e r o a n d O n e

Arabic numerals displaced all others because of their great convenience. They are most convenient to use be cause they give us a way of writing an indefinite amount of

numbers while using only a small number of symbols called digits. This feat is accomplished by attaching different meanings to the same digit. In the number 111, three one's are used, and each has a different meaning. The 1 on the extreme right stands for the number one. The 1 in the

second column from the right stands for the number ten,

and the 1 in the third column stands for the number one

hundred. The symbol stands for the sum of one, ten, and

one hundred. Because the meaning of a digit depends on its position in the written numeral, we say the Arabic sys tem of numerals is a place value system. To represent

three hundreds plus two tens plus five ones, we write 325.

wfputTs'htoThrtp* f®""

won't recognt it «><= But we';

something do^vn in tho « column unless we writel

s a i y t o t h i n k o f t h r e e i t n e c e s - '

to introduce a symbol tn r ^ ^

«se the symbol 0 for tbî« ® absence of ones. We

c o n c e p t o f a n u m b e r i t z e r o . T h e

the Hindus, and was later/'if conceived '

buJt into thei systim of

number in the natural numhpr became a new ,

corporated into the addS to be in- i

a way which is consistent with^h ^^^^^Pbcation tables in :

d o n e b y t a b l e s . T h i s ■

_ P i u s a n y n u m b e r i r i v ^ i t h

again; :

an/ rT f ia symbL as foi?" these ,

any natural number a; In utl v 0-f-a;=:r for

number systems^ir^'lTt hnild-r

elemenUhaTLfnTmbet:;" t"'=='=^a.y for

system. In our « I of th/ contain an

^st rule as onr^^'-î ■ ^ element number

«umber a such that a + ir^^ ^ ''"^ber syrtem " f ®

system, then wp ok n , for all , m contains a

to represent the operation. In that case, the characteristic property of 0 could be written in this way: 0 x x. In the same way, we could, if we wished, change the symbol for multiplication. If, temporarily, we used the symbol * to represent multiplication, then the characteristic property

of 1 would be written as follows: 1 * x = x.

The similarity in form of these two statements empha

sizes the fact that 0 and 1 really both have the same prop

erty, except that each has it in relation to another opera tion. They are both examples of what is known as an identity element. In any system in which a binary operation is defined, and is symbolized by *, if there is an element e that has the property e * x = x, for all values x in

the-system, then e is called an identity element. The letter e is used in this definition of an identity element because it is

the initial letter of the German word einheit, which means unity..

Now we can state more precisely how the terms zero

element and unity element are used in mathematics today. Whenever a binary operation is denoted by the symbol-h and is called "plus," the identity element for that operation

is called a zero element, and is denoted by 0. Whenever a

binary operation is denoted by the sjonbol -, and is called "times," the identity element for that operation is called a

unity element and is denoted by 1. We shall use this con

vention many times in later chapters.

Points on a Line

It is possible to represent the natural numbers as points on a line. On any straight line, choose a point and call it 0, This point divides the line into two half-lines. On one of these half-lines, choose another point and call it 1. Now continue locating points further and further away from 0 by making the distance from each point to the next one the same length as the distance from 0 to 1. Label these

new points successively 2, 3, i, 5, etc. We then have an

endli^ sequence o£ points that is in one-to-one corre

_{spondence With the natural number system. The number}

^ 0 . e x p r e s s e d i n

tei™ of the distance from 0 to 1 as the unit of length.

We can define addition and multiplication for these

•pomts by means of geometric constructions. Here, for

ex-ample, « one way of domg it: To add a and b, meiure out

from a in the toection away from 0, a length equal to

the distance from 0 to b. The point located in this way

has a distance from 0 equal to a + 6. To multiply a^d 6,

Ungtfib

Imgthb

first draw another line întersectînrp '

same scheme for assigning minnk ® at 0 TTor. *ha. i

Locate points 1', 2-, 3', etlfoTjc^,?» Points on tS I

points are separated by en»»? j- so thaf J •

distance from 0 to 1'. ^ftances, aU ,

original line. Locate 6' on thrt ® line '

q equal to b. Then througj atTLf/

-line just drawn from r fn r a -line _ from ;

at a point that will represent cross thp^^- ? '

The construction for oS .® * lino i o r d i n a r y a d d i t i o n o f n u r ? u o b v i n n d ' plication corresponds

tcason: If we designateV'^^aiy ault rtf''?.'^ for multi- !

have defined :

/

as the product of a and b, then x is its distance from 0. The

triangles (0 1' a) and (0 b' x) are similar, so their corre

sponding sides are proportional. Then 1 : b = o : x. From this proportion, we find that x = a • b. With addition and

multiplication defined by these constructions, the system of points on the half-line is isomorphic to the natural

number system.

After we have assigned numbers in this way to points on a line, we find that there are still many points on the line

that do not have numbers. All of our numbers are on one

t h l t

^

b e t w e e n

t h o s e

fteîe3Tn° numbers. For example,

tween^nnH 1 *° the points that lie

be-by steo as wp ' f that will be remedied step

eystenr and tire whole toe^w^'L^pMc"'''

The Natural Nmnbers

thft'a^rcaMbWr"°°™^r'' systems

syateS coSs o "^tural numbed One

their adS and m^lt^^SL^Î,"

consists of the Rnmnrt tables. A second one

third systl'tnfX:?prifS'atp ^he

ate constructions for addinc or t!^r

appropri-vanety of representations raises th?n This

narrai number system, anyhow?" ^ ^

^ we bave defcred the S f -"«nber s^te,L

ye mtend to produce som^'mmb^^ "

■

'="= larv^ S

interchangeable with the natSn? that are not

cha;acttls^^^p."l!?^?.Weni. -t all. To

mon. We must iin+ ^^P^^csentatmta ? merely

acteristics. This is dnn^K''''''"^^'''y tts disf ^ ^

ietica in such a w^the

char-eharacteristics must systems ta

eharacter-eeleotion of cha?i:ife,t&'Phic to^aoM'''^^^ these

only one structure is csT, j"®' ^Sectivelv Hpfi^""- ^"''b a

structure. Here is a systpm system o^ ^ ^''OB one and

^system (not mcSy^ofaxiomsV.^ for the

matician Peauo; S first formulai aMm- j

3 2 ^ ^ t ^ ' t b y t h e m a t h e - !

A set of elements is called a natural number system if it has the following characteristics:

(1) It contains an element called 1.

(2) For every member in the system, there is another

member (and only one) called its successor. (3) Two distinct members do not have the same suc

c e s s o r .

(4) There is no member of the system that has 1 as its

s u c c e s s o r .

(5) If a set of elements belonging to the system contains 1, and, for each member that it contains, also con

tains its successor, then this set contains the whole

system.

Notice that addition and multiplication are not men tioned in these axioms at all. Peano defined these operations in terms of his axioms as follows: For any natural numbers

X and y,

let X +1 = the successor of x;

let X + (the successor of y) = the successor of (x -}- y)f;

let X *1 = x;

let X ♦ (the successor of j/) = x • y + ».

With these definitions it is possible to prove that the natural number system obeys the five laws.

What Peano did for the natural number system is typical

of the way in which mathematical structures are studied

today. In modern mathematics, a mathematical structure is often defined as a set of objects that satisfies a specified set of axioms. If the structure defined is to be unique, the axioms are chosen so that all systems that satisfy the axioms will be isomorphic to each other. Different sets

of axioms have been formulated for the various mathemati

cal structures needed in practical applications.

D O r r Y O U R S E L F

1. By using double-headed arrows, as on page 26, set up

a one-to-one correspondence between the numbers 1, 2, 3 3

3,4, 5 and tîie letters a, e, i, o, u.

2. An addition operation for the system consisting of two elements, a and b, is defined by the following table;

+ a b

a a b

b h a

a) Does this sjratem have a zero element? ;

b) Show that addition is commutative in this system. '

c) Verify from the table ihat (a-f 6) = (o-f-a) j

+ b . I

3. Let the symbol M stand for the binary operation "take ' the maximum of." For example, 5M7 means 7- 8 M 3 means 8; 6 M 6 means 6. Compare 8 M 3 with's M 8 •

If a and.b are any two natural numbers comnnr« nUh

with b ilf o. Is the operation M cnmmnf +• ^ a M b

SMiZM7) with (sV 3m 7.

are any three natural numbers P^T«r^ L ' ' ^ ^ '

*

■

5 . 5

' »

• ! -

«

■

associative law, then x + («V «Peration obeys an

v^ues of X, V and c. ^ + ï) + s for all

_{' wa; rXl d^ht - an abbreviated}

sS rr ^ ^ « ort'rrrth"''""f

b) Prove the Associative i . ' "S"

from the table that aU^'^hTs^eSS^ar^^r^^S

3 4

CHAPTER II

Mmier Systems viithmt

"Numbers"

the word "omte

ormas^E-refers to a symbol in the definition of a

We have broken away , Jf i We defined a nunaber

number system given m two binary operatio]^

system as any set of objecte listed on pages 24-5. In

L defined that obey the five

meaaur-this definition, there is no , _jth the way in which

ing. The five laws are concerned y addition and

the numbers are related to each . separated multiplication tables. To ^^«t of cardinal

num-the concept of natural number fr of actual sets

ber. While cardinal nimbers counting, natur^

of objects, and are intrinsicaBy re . meaning lies m

numbers abstract sj^bols

the formal rules by whidi we convey the full Nevertheless, the natural n^b definition meaning of our break numbers effected a of number. Introducing the not a divorce,

separation from the cardinal rVme in the

back-The cardinal number system w s natural nunaber

ground, because it is iaomorpoi® . . ^ ^^^t no

wg^-system. This fact may arouse the su^

intro-cant change in the concept of gggentially bound up

duced, and that numbers^ °^®TTr.wPver a real change

with counting and nf number system. The

1

S^°bv°nr^j^ chapter is to demonstrate this fact

convino-hers " Thf» î"™!? number systems without

"num-SO are not "numK » • counting or measuring, and

coConly use^ HoC

eystems in the sense ofTu^'diStbm

Subsets of a Set

of tteÏÏpÏÏoS

objects. The objects that hf{n f ^ ^ coUection of

elements. A set is a to a set are called its

elements of the set Thi^ m which objects are

by which the eleinents stating some rule

putting the elements on dis^L Th '

used for a set is a n«i». u . symbol commonly

the set on display inside or w>Wvf ' elements of

Identified prtoed Se' «ley are

b y a m i e : i a a s e t d e L e d

{natural numbers larger than 4, but less than 101

lue same set can be represented by nuttino. v •.

d i s p l a y : ^ i t s e l e m e n t s o n |5»6, 7,8, 9}

Bome deme^°™or et^nf by removing

SrseU6%T«lon^ aboTe """^«.«le

ele-include what is **> extend thé notion f'

will be a "set" witTt i remove all fK i ^

t o a s t h e " t ' o V L t I t

of braces with no elemprt^ We sh^U u®

by removing none or display insi^ a ^ Puir

ft is caUed a «ifeTpf®®'nU of the e^ obtained

_{{==- has eighttbslV ^ ®rr'''Siven}

30 Psets, hated below: the set

{a:,y,z} {a;, 3/} {x,z} {y,z] { x } { y } { 2 } I )

Notice that the given set is one of its own subsets, and th©

empty set is one of the subsets, too. Operations on Subsets

To define a number system, we must first specify what the elements of the number system are. We shall use as

elements all the subsets of a given set. As a specific ex^ple,

let us build a number system out of the subsets of the set

y> z}. For convenience in talking about them, let us

assign a name to each of these subsets. We shall use capital

letters for their names, as follows:

J = {x, y, z} D ~ {a:} A= {x,y} JB = lî/î B = {x,z] F = [z} C = {y,zî 0 = { Ï

The symbols I and 0 are included among the names used

for reasons that will become clear later. ^

The next step is to define two binary operations on these elements. A binary operation is defined when we set up some rule for assigning to each ordered pair of subsets some

particular subset in the same list. We define the opera ion

of fornfing a union of two subsets by means of this ^le;

The union of two subsets is another subset formed y a g as its elements those elements that are in one or m other oi

the subsets being united. For example, A contains tne elements x and y. B contains the elements x and z. ine

elements that are in one or the other are x, y, and z. oo xne

nnion of A and B is the set [x,y,z}, which we have called 1, The union operation will be the "addition , this number system. However, we shall ^ sign to represent it. Instead, we shall use the ,*

The union of A and B will be written as A U f ,

as "A union B," We have seen that A U B = I. The method

of fin(^g the union of two subsets will be clear from the

loliowing examples;

i" C = {x,y,z]\J [y, z} = [x, y,z} ~I DKJE ■= {x) U (yj = {x,y] = ^

CUO= {y,z]\J{ } = [y,z\ =C

The results of forming all possible unions can be

siun-niarized in this table of unions (the addition table for the numbw system we are constructing) :

U I A B C D E F 0

, The second binary operation we define is that of ^0^' *og the intersection of two subsets. The intersection of

two subsets is another subset formed by taking as it®

ments all those elements that are in both of the sub^iPtS being

mtereected. For example, A contains the elements x y. B contains the elements x and a Onlv fiT i ? îo L both A and B. So the intersectinn f ? ® element X is in

which we have%Tlïrn^ Is the subset

be the multipHeation" opêrÏt ^^tion operation

r - " -

' •

The intersection of A and B will be written as X r> B, and

is read as "A intersection B" Then we see that A r\ B = D, When two subsets have no elements in common, their inter

section is the empty set. The method of finding intersections

is shown in the foUowing examples:

znc = {x,y,z]r\ {y,z] « [y,z] = 0

A r \ D ^ { ^ , y } A _{{ x } = { x }}

B n o = {x,z\ r \ _{{ 1 = 1 }} = 0 E r \ F = {y} A _{{^1 = { 1} = 0

The results of forming all possible intersections can be sum

marized in this table of intersections (the multiplication table for the number system we are constructing) ;

A I A B C D E F 0 / I A B c D E F 0 A A A D E D E 0 0 B B D B F D 0 F 0 C C E F C 0 E F 0 D D D D 0 E 0 0 0 E E E 0 E 0 E 0 0 F F 0 F F 0 0 F 0 0 0 0 0 0 0 0 0 0

Tbe Five Laws Are Obeyed

The operations "union" and "intersection" obey the

five laws listed on pages 24-5. We can verify this fact by

referring back to the meaning of these operations. Let us ex-s-nfine the laws one at a time, to see if they are obeyed.

3 9 B C D E F B B B C C c B E B D C E C E B C B F B D E F

Der system. Similar t», u intersection f «"Dsets of

from the subsets of anv^ can he"""®

opLtitrr'"''"^^"' ««"• Union is our addition

^eration, so we must see whether X\JY = Y\JX wliere

_{F represent any subsets of /. X U F La}

that co^ists of those elements that are in X or T y Jx

F or'x The\1 «f those elements that are iii |

^ Seyrf ' ^ets, so law number 1 !

(X U ^ uz"='x uT/u 7?°^?; whether

the set eonsifting rfe^Lnts : ^ U) ^ ^

set X U (F w 7^ io +v x ^ in F.

are in X, or in F or 7 t?® consisting of elements that

60 that law number 2 oheyed^^ obviously the same sets,

XnF = yf^jç- X r\ Y r^' must see whether

ments that are in both X and^xV^ v^ consisting of

ele-s^ting of elements that nr^ • V ^0 means the set

con-obviously the saL set so W ^ These sxe

The oesoaaae tw

means the set consistrng of (-X" H F) H ^

aod also in Z. x n fv rf 7! in and F,

clei^nts that are in X L consisting of

aSi" Ï " ff AS iï

f "5 SlJS)*?'" 0- u z)

■

?£f=?-dB?c?"A"«

we started with a different number of elements, we would

have obtained a number system with a different num er of members. For example, a set with two elements nos rour

subsets. A set with four elements has sixteen subsets. A set with five elements has thirty-two subsets. In gener ,

a set with n elements has 2" subsets.

Zero and Unity Elements

The number system we have constructed has a zero ele

ment and a unity element. Since union is our addition oper

ation, a zero element would have to have the proper y a

"When it is united with any element of the syst^, that element unchanged. A glance at the union a c page 38 shows that the empty set has this property, mat is why we used the symbol 0 to represent it. Since intersec tion is our multiplication operation, a unity element wou have to have the property that when it is intersecte wi any element of the system, it leaves that elemen un changed. A glance at the intersection table on

shows that the original set / has this property. We chose the symbol I to represent it because of its resemblance

number 1,

Special Properties

The number system we have just constructed out of t^e

subsets of {x, y, z} has, as we have seen, some properties

that it shares with the natural number system. T ese in

dude obedience to the five laws, and pos^ssion of a z^o

element and a unity element. However, it also has 6® peculiar properties that are entirely unlike the proper 1 of the natural number system. A few of these are uo e

h e r e . _ .

1* We can see from the tables that for

the system, X U X = X, and X O X = X That is, a suV

set united with itself yields the same subset, and a intersected with itself yields the same subset. In t en

[ierule.O + 0 = 0, but 2 + 2 is not 2.1 - 1 = 1. but 2 • 2'

i n o t 2 . '

2. We have already observed that intersection is

distribu-ive with respect to union. It can also be verified that union

i distributive with respect to intersection. That is, in the a ement of the distributive law, union and intersection

an c ange places. This, too, is unlike what we found in the

latural number system. There, while multiplication is

^^ibutive with respect to addition, addition is not

dis-nbutive with respect to multiplication.

mp fKflt in the system, we can find another

ot wi elements that the first one does

me* hf.nfln subset the complement of the first

0 )

S

( t h e i r

i n t e r s e e t i o n

s n' they complete the original set (their union

JemeS^t. 'î'"' ty^tem. we denote its

coin-^ot ™bsets of {x, y, x}, A =

r-Vo TU ^ =W=^'- Similarly. jB' = E, = B and

bllowing proSSr "taking the complement" has the

X n X' == 0, X u Z' « 1; (XO' = X

(zuy)' = 2:'ny';(rr>y)' = x'ur

The truth of the law can be observorî v

cons^ts of elements in one set or that a union

ment of elements in one set and intersection

Bame aa 'Ct h, |f? "»^t "not StW z"'"

and Y" is the ^ ^^t in K». «y. j îv is the

Bhow tha?™:,?;;^.°thiZorhi

42 statements are corrit. thought

T h e A l g e b r a o f L o g i c . v +

The number system we

is only one of a whole farmly of ^ nltrebras The

similar properties. They are called Booto

type of structure that they +^rLtical application

matical curiosity. It has an important p ip-tronic

com-in the study of logic, and com-in the d^ign

puters. In logic we study relationships out in

The analysis of these relationships can . .

state-symbols in the following way: Let each prop ^

ment be represented by a letter, j *vp symbol

symbol U for "or," the symbol n for '[and/ md tbe s^bol

' for "not," as we already bave done m statement

Use 0 for any statement that is false, and I , . „gj,^ and

that is true. With this notation, the class their logical relations becomes a Boolean

algebr^ are named after tbe English mat e , - '

Boole, who pioneered in the study of symbo gi

DO IT ÎOXJRSELF

1. Assign names to the subsets of the set ^f'^L^g^mpty

1= (a;,y],A= lx},B- lî/},0= n

a) ctiistruct a table for the union operation for this

s y s t e m o f s u b s e t s . . ^ n n e r a t i o n .

b) Construct a table for the intersect on operation.

2. Let I represent the set {a, h, c, d, *' Let X represent the subset [a,

Let y represent the subset la, 6, „x ^.e x in I?

a) What elements are in X', the comp ement of A m

b) What elements are in Y', the complement of T m

c) What elements are in X' E *

d) What elements are in X

e) What elements are in (X H E) ' show that f) Compare your answers to c) and e; w

X ' U y = ( X n Y ) ' . h r d l .

3. List all the sixteen subsets of the se 1 j j '

C H A P T E R i n

JVumbers from Old

Questi

o^ nat Have No Answers

« d e s J ^ 6 n a t u r a l n u m b e r s , b e

-bave occasion to subtract sometimes

tion can be defined in operation of

subtrac-'eally asks us Z Symbol 5 - 3

2 Since the n&tm&l number added to

2. we say 5 — 3 = 2 w« nTi 4.Î question is the number

tween 5 and 3. The question o. difference

be-of an equation, a: + 3 - g „ written in the form

to the equation ^

^pts us to tty to th^ between 5 and 3

numbers chosen at r between any two

&

i n t o

the B^h written as 3 -L g the difference a i d e d t o 5 1 ' ' m a t

isn't fln,v « V ' Unfortunatelv natural number

subttahSd''i^^;^'* t£ 'bbe

"What natural number «^5 of it ao ?i, ^ doesn't

whi^r answer lY ^ 0?» fiî question,

which ^ the same qL/. '^se the eau«r 't doesn't

a solution. This is a dpf t^^en it dS« ® + 6 = o,

44 efect of the natural have

number system

that limits its usefulness. Because, although the question

3 — 5 is meaningless for the natural number system, there are practical problems that lead to just such a question. Tor example, if the temperature is 3 degrees, what will it he after the mercury drops 5 degrees? It would be useful to have a number system which contains a number that

can serve as the answer to this question. The defect of the natural number system that we have observed confronts us with a challenge. Can we construct a number system

that does not have this defect? Can we build a number

system in which subtraction is always possible for any pair

of numbers taken in any order, so that an— h always has a meaning, and x + b = a always has a solution? We find

that we can.

Readers who have had high school algebra will remember

that a system of numbers that includes "negative" as well us "positive" numbers is supposed to serve this purpose.

Rut in their course in high school algebra, they were given this system as a finished product obeying certain mysterious rules such as, "the product of two negative numbers is a positive number." In what follows, we do not take the existence of such a number system for granted. We prove it exists by actually constructing it. We also remove the

^ystery surrounding its rules by actually deriving them

from the familiar rules governing the system of natural

numbers.

Families of Diflferences

To construct the improved number system, we use a

rather interesting device. The symbol a — b asks us a ques-does not always have an answer. To make sure that it will have an answer in the new system, we let the

question be its own answer I In effect we say, let each expr^sion like 5 — 3, or 3 — 5, or 2 — 7, represent a

num-^ system. To justify calling these strange

things numbers we shall have to define addition and multi

plication operations for them, and then show that with these operations they really constitute a number system.

However, we run into some complications even before we

take our first step in this direction. In the natural number

system, 5 — 3 does have an answer, and the answer is 2. But 2 — 0,3 — 1,4 — 2,6 — 4, and an endless list of sunils'

symbols also represent 2. So we cannot simply let each si^

symbol stand for a separate number in the new system. " ®

would want all of these symbols to represent the same num

ber, just^ they do in the natural number system. We take

care of this diflSculty by using as the elements of our neW

number system, not single symbols written in the form

between two natural numbers, but whole

Terences. The first step is to estabh^

hplontr 7 03,n recognize when two such symbole

sSd f^y. We get a clue to the rule «8

seXh! by examinmg the difference symbols that

repre-6 - 4 r e n r f . v ^ t h e d i f f e r e n c e

the left mmîî f number. Notice that if we add

Step, pirgt ^ our construction step by

numbers, such as 7 L!Î k ordered pairs of

ye write the "diffprprt ^ 9> 15 and 1, and so on. The

m a definite order «5*^ numbers in the pair, taken

bave a meanine in difference does not alway nse an ordinarv inîmi="^ number system, we shall no

the symbol — instenri ^ben we write it. We shall us

subtraction of natumi' '®rnind us that this is really no

Bested by subtraction but merely a symbol

sug-^ 1. So now have symbols like 7 - 5,

d^erence" between nah will be called »

associaS lumbers.

to ^ tbe foSj^^ difference a whole family

/ o r T / i e

_{^ - u + Hn difjerïnces «}

■

_{- ^}

to a differenced^-designate the faimly that

be-4 6 w r i t e t h a t d i f f e r e n c e i n s i d e

parentheses. Thus {a ~ b) means the fa^y

that belongs to a ~ b. The sjmbol (3-1)

ily of differences thatbelongstoS - Ï. bLau^

that the difference 6 - 4 belongs to

3 + 4 = 64-1. We call these famihes of differences îî>

tegers. They will be the elements of our new number

sya-^We observe immediately two characteristics of these

families which we call integers: ., t? - û-tromnle

1) A difference belongs to its own fa^y. Tor oxamp ,

3 -1 belongs to (3-1). This follows from the fact to

uu belongs to (a —b) ^ 'oii In

een-a = 3,b = l,« = 3,een-andu = l,een-and34-l = 3 + l-togen

eral, a - b belongs to (a - b) because a 4- 0- uj •

2) If one of two differences belongs to the family

other, then toey have the same families, suppose, m

ample, that a - b belongs to toe fan^y to

we can show that every member of (a - b) ,

(c d), and vice versa. If P9 belongs to ^

by the criterion for membership in a faimly, a + 9

However, a —b belongs to (c —d), "Y a + b

Adding these two equalities, we get a + b y e + 9 . ^ + p + d. Taking away a + b from both « ^e-= P + d. But this is equivalent to saymg that p

longs to (c d), according to our criterion ^ ^ in a family. This shows that any member ot ^

belongs to (c d). A similar argument, gomg ,

same chain of steps in reverse, shows that ^ (c~d) also belongs to (a~b). So the fore

and (c — d) have the same memberships* and

'^Th^econd characteristic of these f^es ee^'l

h a s t h e s e c o n s e q u e n c e s : F i r s t , e a c h b e

to one and only one mteger. Secon<By, an g ^

represented by putting on display P ^ ^ 2), one of the differences that belong to it. bo ' ^Yie

cri-( 5 3 ) a l l r e p r e s e n t t h e s a m e ^ ^ t e s t

for equality of integers. That is, the integers (a h) and

(c ~ d) are equal if and only if a + d = c + b. For exam ple, to prove that (3 ~ 1) = (4 2), it is enough to ob* ;

serve that 3 + 2 = 4 + 1.

Addition and MuItipKcation of Integers

Now we define addition and multiplication for the system i

ot mtegers. We assign a sum to any ordered pair of integers

by means of the following defining equation:

(a - b) + (c ^ (i) = (a + c ~ 6 + d)

"de of this equation

repre-- . . because, if a and c are natural numbers?

number "xhen S^Uarly, b + i is a natural

bers and tVtc. ^ ^ ^ is a difference of natural

num-W.^i- that belongs to it.

means of thi ? to any ordered pair of integers by

means of the foUowmg defining equation:

( a

b) • (c - d) = (a . c + b . d o • d + h • c)

Sr sy^bof Lside\ht"^ side represents an integer, because

o f ^ d i f f e r e n c e

nuS.eï T

difference to represent each of thp int ' another

_{will we stiU get the same^^? ff Jf h adding,}

m useless. However, the de^ltio^''" our definition

^Uow the directions it gives for fi ^^osen. If we

oers of the mtegers are used tf Matter which

mem-;«?S t •" "S,f ,2;« a™ "n

(5 3) _}_ (65) = (5 + 6 3 + 5) = (11 8). However, (5 3) could also be represented by (4 — 2), because 5 + 2 = 4 + 3. Similarly, (6 5) could also be represented by (5 4), because 6 + 4 = 5 + 5. If we apply

our definition to these other representatives of the two

integers, we find that

(4 - 2) + (5 ~ 4) = (4 + 5 2 + 4) = (9 - 6). By using different representatives for the two integers we were adding, we got sums that look different. However, al though they look different, the sums are the same.

(11 — 8) = (9 ~ 6), because 11 + 6 = 9 + 8.

^ The same problem arises in connection with our defini tion for multiplication of integers. The definition makes use

of a particular difference that belongs to each integer. But it can be shown that it does not matter which difference

that belongs to an integer is chosen as its representative. They all lead to the same product anyhow. So there is no

ûmbiguity in our definitions of addition and multiplication.

Tbe Integers Form a Number System

We now have a system of elements called integers, with

an addition operation and a multiplication operation de

fined for this system. To show that the integers form a mimber system, we have to prove that the operations obey the five laws listed on pages 24-5. As an example of how such

®' P^oof is carried out, we give the details of the proof for

the commutative law of addition. Let (a —' b) be any

in-^gar, and (c ~ d) any other integer. We must show that

7 îï) + (c d) = (c - d) + (a - b). Applying our def-mition of addition of integers, we find that (a b) + (c d) = (a + c b + d), while (c d) + (a b) = (c + a d + b). But natural numbers obey the

commuta-^ve law for addition, so a + c = c + a, and b d = d-\-h,

the same integer. Therefore, (a —' b) + (c — d) =

(c d) + (a ~ b), and the commutative law for addition 4 9