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I have proved that between every $2$ distinct real numbers, there exists infinitely many other real numbers. A question came up to my mind: Now that we have an infinite number of real numbers between let's say, $0$ and $1$, can we represent all real numbers as numbers between $0$ and $1$ too? In other words, can the set of all real numbers be put in one-to-one correspondence with $[0,1]$? If we can, why? And is there a way to "choose" the number which shows the real number that we wanted to represent it as a number between $0$ and $1$? For example, how can we find the number in $[0,1]$ that represents $\pi$?

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Yes, you can. There exixts a real number whose binary representation begins with the complete texts of the Library of Congress, say, in ASCII, followed by the same as 600 dpi PDF scans, followed by how Shakespeare would have written Romeo and Juliet if he had been born Tanzania in 1950. And then some (in fact a rational yould suffice for any finite requirement). – Hagen von Eitzen Nov 15 '13 at 14:53
you still count them with numbers great than 0 and 1. I think it's kind of like saying that everything between you and me can be represented by you and me. – whybeing Nov 15 '13 at 14:59
Are you saying that $\frac12$ is greater than $0$ and $1,$ for example? – Cameron Buie Nov 15 '13 at 15:01
This is not so bad a question as to be closed as off-topic in my opinion, at least if we take it as a "popular math" kind of question, but I would be really surprised if this is not a duplicate... Not sure how to find it, though. – tomasz Nov 15 '13 at 16:01
$\tan(x)$ also works (or rather, it works for $[-\pi/2, \pi/2]$, but a quick argument gives you a bijection $[0, 1]\rightarrow [-\pi/2, \pi/2]$). – user1729 Nov 16 '13 at 14:21

In the sense that there are only finitely-many "things" in the world, absolutely. In fact, we can represent each thing in the world as a number of the form $\frac{1}{n+1}$ for some positive integer $n$.

However, there are still a prohibitively large number of things in the world, so actually choosing such representation is impossible in practice, though possible in theory. (And if you count electrons as "things," then it's impossible even with unlimited time.)

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Thanks for the note, I added another question of "choosing" them as you stated here. – CODE Nov 15 '13 at 14:56

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