Page 45 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

because the integers are closed under multiplications means that ac ∈ Z , and the natural numbers are

also closed under multiplication which means that bd ∈ N. So by definition xy is the product of ac
with the multiplicative inverse of bd , and xy ∈ Q .

To show that Q is closed under addition and subtraction we can put the multiplicative identity in
disguise. Since d ∈ N , d =0, and the multiplicative inverse of d is in Q , so

1 d 1 d
1= d ∗ = ∗ = .
d 1 d d

Since

x =1 ∗ x,

by the multiplication rule for the rational numbers,

d a ad
x = = .
d b bd

Now we can put 1 into a different disguise so that he can put his friend y also in disguise. Since b ∈ N,
b =0, and the multiplicative inverse of b is in Q so

1 b 1 b
1= b ∗ = ∗ = .
b 1 b b

Since

y =1 ∗ y,

by the multiplication rule for the rational numbers,

b c bc
y = = .
b d bd

So, by the addition and multiplication rules for rational numbers,

ad + bc ad − bc
x + y = , x − y = .
bd bd

Because b and d are in N , and the natural numbers are closed under multiplication,

bd ∈ N.

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