Introducing Set Theory: Proving Real #s Identical in Bases

[Blackberg]

I'm introducing myself to set theory. My reference doesn't seem to address the fact that 1/1 = 2/2 = 1. If we make a correspondence between natural numbers and rational numbers using sequential fractions, should we just skip equivalent fractions so as to make it a bijection? In other words, does it matter whether the correspondance is injective or not, or whether it is surjective or not?

I'm also wondering how one would prove that the set of real numbers in base ten is identical to the set of real numbers in another base.

 

[mfb]

If you want to make a bijection it is important. If you just want to show there are not more rational than natural numbers, it does not matter.
The reverse statement (there are not more natural than rational numbers) is trivial anyway.

I'm also wondering how one would prove that the set of real numbers in base ten is identical to the set of real numbers in another base.

Numbers do not have a base. You can express a real number in a specific base to write it down, but the number itself is independent of it.

 

[FactChecker]

If you are looking for a simple, clean proof, I don't think there is one. It could be done, but it might be a lot of work. You can define the mapping between the real numbers and their representation in any base. Composing the mappings should give you a mapping between the two representations. There might be a lot of tedious complications with infinite length representations like 1 = 0.9999999...are two base 10 representations of 1. Irrational numbers are another complication.