Page 41 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

Is an integer? No. So, the integers do not have a multiplication swingset. We need a bigger playground.
1
2

Definition 3.2.10 A rational number is one of the following:

1. An integer;
2. The multiplicative inverse of a natural number, which for n ∈ N is written as

1
;
n
3. The product of an integer with the multiplicative inverse of a natural number. For z ∈ Z and
n ∈ N the product of z with the multiplicative inverse of n is

z
.
n

For every z ∈ Z ,

z
z = .
1

The set of all rational numbers is written Q.

You may have seen the rational numbers defined differently like in the following proposition. As long as
we can prove that two definitions are equivalent, then they have the same meaning, and so mathematically
it is equivalent no matter which definition we choose to use.

Proposition 3.2.11 (Rational Proposition) Any rational number can be written as

p
q

where p ∈ Z and q ∈ N .

Proof: If a rational number is an integer, then that integer is the “ ” in the theorem, and since
p
p
= p,
1

the multiplicative identity 1 is the “q ” in the theorem. If a rational number is the multiplicative inverse
of a natural number n, then it is

1
,
n

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