Next, consider the real numbers, denoted by ℝ, as a line extending infinitely far in both directions. In this book, the notation, ≡ indicates something is being defined. Thus the integers are defined as

the natural numbers,

and the rational numbers, defined as the numbers which are the quotient of two integers.

are each subsets of ℝ as indicated in the following picture.
As shown in the picture,
Axiom 1.3.2 x + 0 = x, (additive identity).
Axiom 1.3.3 For each x ∈ ℝ, there exists −x ∈ ℝ such that x +
Axiom 1.3.4
Axiom 1.3.5 xy = yx,(commutative law for multiplication).
Axiom 1.3.6
Axiom 1.3.7 1x = x,(multiplicative identity).
Axiom 1.3.8 For each x≠0, there exists x^{−1} such that xx^{−1} = 1.(existence of multiplicative inverse).
Axiom 1.3.9 x
These axioms are known as the field axioms and any set (there are many others besides ℝ) which has two such operations satisfying the above axioms is called a field. Division and subtraction are defined in the usual way by x − y ≡ x +
Here is a little proposition which derives some familiar facts.
Proposition 1.3.10 0 and 1 are unique. Also −x is unique and x^{−1} is unique. Furthermore, 0x = x0 = 0 and −x =
Proof: Suppose 0^{′} is another additive identity. Then

Thus 0 is unique. Say 1^{′} is another multiplicative identity. Then

Now suppose y acts like the additive inverse of x. Then

Finally,

and so

Finally

and so by uniqueness of the additive inverse,