Page 65 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

• Hint for # 9: Let’s show that there is no rational number that is closest to and bigger than 77.
The best way to do this is by contradiction. Assume there is a rational number that is closest
to but not equal to 77. We don’t know what it is, but we do know it’s rational. So let’s call
p
this mystery number x. Then, x = for some integer p ∈ Z and q ∈ N . We know that
q
p
x = > 77.
q

This means that

p
− 77 > 0,
q

and so

p − 77q
> 0,
q

and since q ∈ N

p − 77q> 0.

Since p , q , and 77 are all integers,

p − 77q ∈ N.

So,

p − 77q ≥ 1.

Dividing by q ,

p 1
− 77 ≥ ,
q q

which we can re-arrange to

p 1
≥ 77 + .
q q

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