Rational Numbers

the structure + , X , and < ) of the integers as first constructed to those described by pairs and equality. In either construction, the integers

Z

can be described as the minimal way, unique up to isomorphism, of embed

ding N in a larger structure in which subtraction is always possible, preserving all the algebraic properties of + ^and X . Here as elsewhere, what matters is not an exact description of what an integer

is,

but a description of the structure of all the integers, up to isomorphism.

To keep accounts, one often needs to divide numbers evenly into parts

and this often cannot be accomplished with whole numbers. Fractions provide the answer. They are introduced individually, as

1 /2, 2/3, 1 /5, 4/5,

etc., and then manipulated in the evident way :

m m ' mn '

m 'n

- +

n

n ' -

nn ' ' m m '

n n ' mm '

nn ' . ( I )

This practical process suggests a corresponding formalization. The ini

tial observation is that division is not always possible within the set N + of all positive natural numbers. Hence one is led to introduce the set Q + of all ordered pairs (

m,n

) ^of

positive

natural numbers, defining addition and multiplication by the evident translations

(

m,n

₎+ (

m ',n '

) = (

mn '

nm ', nn '

), (

m,n

)(

m ',n '

) ⁼(

mm ',nn '

)

(2)

of the practical rules

( I )

and taking care to define the equality of (

m,n

₎

and

(r,s

) by

ms

⁼

nr.

When

m

in N + is identified with the pair (

m, l

₎this again defines a minimal expansion of the set N + to a larger set in which division is always possible and in which all the rules of arithmetic still hold. As before, there is nothing unique about the formulation of this construction. Instead, one might have used only those pairs (

m,n

₎in which

m

and

n

have no common factor (except

I);

in that case, one must modify addition and multiplication in

( I )

to reduce each answer to lowest terms.

With this inconvenience, the "artificial" definition of equality of pairs is avoided. Again, what matters is only the resulting structure up to isomor

phism.

The system Q of

all

rationals may then be obtained from Q + , the posi

tive rationals, by simply adjoining zero and negative rationals. Alterna

tively, one may construct Q directly from

Z

by using all pairs

(a,b )

of integers

a,b

Z,

with the same addition and multiplication as in (2)-and the important proviso that the "denominator"

b

is never

O.

As in previous cases, what matters is not the explicit definition of a rational, but the resulting structure.

5

2 I I . From Whole Num bers to Rational N u m bers

7 . Congruence

A typical clock runs up to the figure of 1 2 hours and then repeats, but one can still do arithmetic on the limited list of hours: Seven hours after nine o'clock is four o'clock. Similarly, in the decimal system there are only ten digits 0, 1 ,2 , . . . , 9 ; the usual rules for addition and multiplication, ignor

ing the carryover to the tens' place, work perfectly well for the manipula

tion of these digits by themselves:

6

+ ⁷⁼^3,

8

+ 7 ⁼

5,

8 · 3 =

4,

3 · 9 ⁼ 7 .

These rules ignore all the multiples of 1 0 ; in a sexagesimal system there are similar rules which ignore all the multiples of sixty. "Casting out nines" is a rule for checking arithmetic calculations. This rule for checking a multiplication says: Add up the digits of each factor, multiply the result

ing sums, and check this against the digit sum of the original purported answer. Thus 32 times 27 calculates to

864.

To check, 32 becomes

5,

27 becomes 9,

5

times 9 is 45, with digit sum 9. This checks with the digit sum in the purported answer, which is

8

6

4

⁼ 1

8

with digit sum 9.

What happens here is that 32 is replaced by

5,

casting out the difference which is 27, or three nines. The reason it works is that factors differing by a multiple of nine will have a product differing (at most) by a multiple of 9. In brief, arithmetic operations are valid "casting out 9's" or "modulo 9".

These examples each involve the use of a modulus: 1 2, 1 0, 60, or 9, as the case may be. The general procedure is similar. For integers

a,

b, and any natural number

m

oF 0 as modulus, one writes

a

== b (mod

m),

or says that

a

congruent

to b for the modulus

m,

when the difference

a

- b is a multiple of

form an (abelian) group under addition. Under multiplication, the non-zero remainders modulo a prime

p

also form a group of

p

I

elements. This is not the case for a composite modulus

m,

such as 2 · 3, because there 2 · 3 == 0 (mod

6)

so that neither 2 nor 3 can have a multiplicative inverse

7. Congruence

53

modulo

6.

To get a multiplicative group for such a composite modulus

m,

on must use only those remainders

r

which have no factor in common with the modulus

m.

The number of such remainders is denoted by

cp (m),

while

cp

is called Euler's

cp

-function. For a prime

p

or for integers

m,n

with greatest common divisor l one readily calculates that

cp (p )

⁼

p - I , cp (pk )

⁼

(p _ l )pk - l, cp

(

mn

) =

cp ( m )cp ( n ) . (2)

These formulas provide for a computation of any

cp (m)

from the prime decomposition of

m.

We cite them to emphasize that the formulation of congruence arises both from practice (multiplying hours or digits) and from number theory.

To say that calculations with congruences

are

calculations with the remainders is a bit artificial. Thus modulo

5

one could replace the five remainders

0, 1 , 2, 3, 4

by the remainders

- 2, - 1 , 0, 1 , 2

or by

- 4, - 3, - 2,

I , 0.

Here, as always, mathematicians strive for an invariant formu

lation. Each remainder

r

stands for (and may be replaced by) the

"congruence class"

emr

all

integers

a

with

a

⁼⁼

r

(mod

m).

To add the class

emr

to the class

ems

one may then take any representative

a

emr,

any

b

ems,

add

a

and

b,

and take the class of this sum

a

b

as the sum

emr

ems.

One must then prove that the resulting sum of classes doesn't depend on the representatives

a

and

b

chosen-but this fact is just a restatement of the rule

( I )

for adding two congruences. With this fact established, we see that the collection

Zm

of all these congruence classes

em

forms a system with binary operations of addition and multiplication, and that the function

em

from

Z

Zm ,

as in

carries the addition and multiplication of integers to that of congruence classes. (It is thus a first example of a homomorphism of + and X .) This gives an "invariant" formulation of the calculation with remainders.

Thus we have (at least) three descriptions of the algebra of integers modulo

m:

^As the ordinary integers taken with a new equality, congruence modulo

m;

as the algebra of remainders modulo

m;

or as the algebra of congruence classes, modulo

m.

The last description is the more invariant-and the more sophisticated, since it involves a set whose ele

ments are sets (a collection of congruence classes). However, all three con

structions yield isomorphic results, and the results are useful (and practi

cally indispensable) for the statement of simple number theoretic facts.

For example, for any integer x, one has always

x2 == ° or

I

(mod

4),

x 2 ⁼⁼

0, I ,

4

(mod

8) .

Another problem is that of finding a common solution x for two (simul

taneous) congruences:

54

I I . From W hole Num bers to Rational Num bers x ==

b

( mod

m ),

x ==

c

( mod

n ) .

(3) In this situation, the "Chinese remainder theorem" states that if

m

and

n

have no common factors (except 1 ), there always is a solution, unique modulo the product

mn.

In document Mathematics Form and Function (Page 62-65)