Page 124 - 'Blast_Into_Math

Blast into Math! Mathematical perspectives: all aour mase are melong to us

3. For any non-zero number x , for n ∈ N , x −n is defined to be

1
.
x n
4. Exercise: Using these definitions, prove that for any x =0, for any integers a and b ,

ab
a b
a b
x x = x a+b , (x ) = x .
In baseball, there are four bases: first base, second base, third base, and home base. When you’re playing
baseball, you have a different perspective when you stand on each of the different bases. If you stand
on first base, you’re closest to the first baseman, but if you stand on home base, you’re closest to the
catcher. Things look different from different bases. It is the same in mathematics, except in the game of
mathematics, there are infinitely many bases, because any integer b ∈ N such that b ≥ 2 can be a base.
So, you can imagine that the game of mathematics is played in a great big infinite universe, and there
are infinitely many different bases that you can stand on, and from each different base, you can have a
different mathematical perspective.

Exercise: Prove that the set of all bases has infinitely many elements. Is it countable? Why or why not?
Prove your answer.

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