Page 43 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

If a< 0, then −a ∈ N , and the multiplicative inverse of x is

b −b b
− = = .
−a −a a

Now we will prove that following the rules, we can play the games of addition, subtraction, multiplication,
and division within the playground of rational numbers.

Theorem 3.2.12 (Rational Theorem) Every rational number has an additive inverse which is also a rational
number, and every non-zero rational number has a multiplicative inverse which is also a rational number.
The set of rational numbers is closed under addition, subtraction, and multiplication.

Proof: First we’ll prove that Q contains additive inverses and multiplicative inverses for all non-zero

rational numbers. To do this we’ll use the Rational Proposition. For any rational number x ∈ Q , there
is a ∈ Z and b ∈ N such that

a
x = .
b

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