Page 44 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

Since a ∈ Z ,

−a ∈ Z,

so

−a
∈ Q,
b

and

−a a −a a − a
x + = + = =0.
b b b b

This shows that Q has additive inverses. Now let’s assume x =0, which means a =0. Then, if a> 0,
since a ∈ Z this means that a ∈ N so

b b a b ab
∈ Q, and x ∗ = = =1.
a a b a ab

If a< 0, then −a ∈ N , so by the definition of Q ,

−b
∈ Q,
−a

and

−b b b a b ab
= , x ∗ = = =1.
−a a a b a ab

This shows that Q contains multiplicative inverses for all non-zero rational numbers.

To show that Q is closed under multiplication, let y ∈ Q , and c ∈ Z , d ∈ N such that

c
y = .
d

Then

a c ac
xy = = ∈ Q,
b d bd

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