In which interval is there a larger amount of real numbers, in [4,5] or in [4,12]? Why?

Question

Yesterday, a friend of mine asked me this during our math class. At first I thought it was easy but I wasn't able to give him an answer. Please satisfy my curiosity and help me find an answer!

Answer 1

In general, any 2 non-empty real intervals $[a,b]$ and $[s,t]$ have the same cardinality, since you can explicitly construct a bijection $f:[a,b] \to [s,t]$ as $$ f(x) = s + \frac{x-a}{b-a} \times (t-s) = \frac{x-a}{b-a} t + \frac{b-x}{b-a}s. $$

Answer 2

You will probably not like the answer, but : it all depends of what you call "having a larger amount of real numbers".

You can find a bijection between these sets : $f(x) = 8(x-4) + 4$, so the two sets have the same cardinality.

If you consider the "measure" of the sets (Lebesgue measure), $[4,5]$ has measure $1$ and $[4,12]$ has measure $8$.

If you consider inclusions of subsets, obviously $[4,5]$ is a strict subset of $[4,12]$.

There is not a better way to define "having a larger amoung of real numbers", just numerous ways depending of what you want to do with the notion exactly.