16 1. The Real Numbers

1.3. Integers and Rational Numbers

The need for number systems larger than the natural numbers became apparent

early in mathematical history. We need the number 0 in order to describe the

number of elements in the empty set. The negative numbers are needed to describe

deficits. Also, the operation of subtraction leads to non-positive integers unless

n − m is to be defined only for m n.

Beginning with the system of natural numbers N and its properties derivable

from Peano’s axioms, the system of integers Z can easily be constructed. One sim-

ply adjoins to N a new element called 0 and for each n ∈ N a new element called −n.

Of course, one then has to define addition and multiplication and an order relation

“≤” for this new set Z in a way that is consistent with the existing definitions of

these things for N. When addition and multiplication are defined, we want them to

have the properties that 0+n = n and n+(−n) = 0. It turns out that these require-

ments and the commutative, associative, and distributive laws (described below)

are enough to uniquely determine how addition and multiplication are defined in

Z.

When all of this has been carried out, the new set of numbers Z can be shown

to be a commutative ring, meaning that it satisfies the axioms listed below.

The Commutative Ring of Integers. A binary operation on a set A is a rule

which assigns to each ordered pair (a, b) of elements of A a third element of A.

Definition 1.3.1. A commutative ring is set R with two binary operations, ad-

dition ((a, b) → a + b) and multiplication ((a, b) → ab), that satisfy the following

axioms:

A1. (Commutative Law of Addition) x + y = y + x for all x, y ∈ R.

A2. (Associative Law of Addition) x + (y + z) = (x + y) + z for all x, y, z ∈ R.

A3. (Additive Identity) There is an element 0 ∈ R such that 0 + x = x for all

x ∈ R.

A4. (Additive Inverses) For each x ∈ R, there is an element −x such that x +

(−x) = 0.

M1. (Commutative Law of Multiplication) xy = yx for all x, y ∈ R.

M2. (Associative Law of Multiplication) x(yz) = (xy)z for all x, y, z ∈ R.

M3. (Multiplicative Identity) There is an element 1 ∈ R such that 1 = 0 and

1x = x for all x ∈ R.

D. (Distributive Law) x(y + z) = xy + xz for all x, y, z ∈ R.

A large number of familiar properties of numbers can be proved using these

axioms, and this means that these properties hold in any commutative ring. We

will prove some of these in the examples and exercises.

Example 1.3.2. If F is a commutative ring and x, y, z ∈ F, prove that

(a) x + z = y + z implies x = y;

(b) x · 0 = 0;

(c) (−x)y = −xy.