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Note: I'm not asking about [0,1], I'm explicitly asking about [0.1, 1].

I'm being told [0.1, 1] is not countable. It's clear for me that [0,1] isn't, but it feels less clear for [0.1, 1], and I'll try to explain my understanding:

Essentially the idea is that if I can map every number in [0.1, 1] on a unique whole number it's countable, otherwise it's not. So the idea would be to just remove the 0. and what remains is the unique whole number.

0.123 maps to 123, 0.123456789101112 maps to 123456789101112, 0.1 maps to 1. Since we're already mapping 0.1 to 1 we can't map 1 to it too, so we map 1 to 0. For any example you can write down in it's decimal form, this would work for. However...

An obvious example where this might/does break down is for instance the number pi/4, which has an infinite number of decimals. So we can't map a whole number onto it. Or can we? I don't know.

So why is it allowed for a decimal number to have infinite digits, but a whole number cannot? Or is that just the definition of whole numbers and the "countability" of a set, like the difference between rational and irrational numbers?

The Oddler
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    Note that there's a bijection between $[0.1,1]$ and $[0, 1]$: send $x\in [0.1,1]$ to ${x-0.1\over 0.9}.$ – Noah Schweber Dec 07 '20 at 17:07
  • A positive integer must have finite many digits because we must be able to reach it in a finite number of steps by adding $1$ over and over again beginning with $0$. In each step, the number of digits can increase only by $1$ (and this only if we arrive at a new power of $10$ , but this does not matter for the argument). Therefore, no positive integer can have infinite many digits. – Peter Dec 07 '20 at 17:35

2 Answers2

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It is not countable. The quickest way to see this, if you already accept that $[0,1]$ is uncountable, is with a map $f(x) = \frac{x-0.1}{0.9}$, which is a bijection from $[0.1,1]$ to $[0,1]$.

An obvious example where this might/does break down is for instance the number $\pi/4$, which has an infinite number of decimals. So we can't map a whole number onto it. Or can we? I don't know.

Correct. Or take a number like $\frac19 = 0.\bar1$: what should this map to?

So why is it allowed for a decimal number to have infinite digits, but a whole number cannot? Or is that just the definition of whole numbers and the "countability" of a set, like the difference between rational and irrational numbers?

Informally, a number can have infinitely digits after the decimal point because each new digit makes it more and more precise: $3, 3.1, 3.14, 3.141, \ldots$ is getting closer and closer to some number (perhaps $\pi$). Adding digits to the left of the decimal point, on the other hand, makes the number explode: $1, 21, 321, 9321, \ldots$ isn't approaching any given number, it's shooting off to infinity.

Théophile
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  • Yes, any number with an infinite number of digits doesn't work. That's why I added the second part of the question, "why can't a whole number have an infinite number of digits?". Because 0.1¯ would just map to 1¯. But from the other answer as well, I can live with the explanation that a whole number can't have an infinite number of digits. That sounds fair to me. – The Oddler Dec 07 '20 at 17:14
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    Oh! Your update, explaining how a decimal gets more precise, but a whole number would explode, that really helps giving it a more "intuitive" feel. Thanks a lot! – The Oddler Dec 07 '20 at 17:15
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    @TheOddler On that note, remember how the real numbers are defined: they literally are (equivalence classes of) "increasingly precise approximations." – Noah Schweber Dec 07 '20 at 17:17
  • Interesting, thanks! Also very kind of you to say "remember how..." as if I ever knew that. – The Oddler Dec 07 '20 at 17:25
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    @TheOddler A lot of math consists of remembering things one never knew. :P ("In retrospect that definition is exactly the thing that formalizes the intuition I had but never actually said out loud!") – Noah Schweber Dec 07 '20 at 17:33
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A simpler example than $\frac\pi4$ is $\frac13=0.3333333\ldots$

Anyway, our way of representing natural numbers is defined in such a way that any such number can be represented using only finitely many digits. And therefore that function that you mentioned is undefined for infinitely many elements of $[0.1,1]$.

José Carlos Santos
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  • Right, this was what I was thinking too then. Essentially it's just a definition. A whole number can't have an infinite number of digits. – The Oddler Dec 07 '20 at 17:11
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    @TheOddler Note that in some sense that's tautological: since "whole number" and "finite" are pretty much the same concept, this is similar to saying "Every whole number has a whole number of digits," which is much less interesting. – Noah Schweber Dec 07 '20 at 17:11
  • Fair, but anything with infinity gets weird, so I guess I'll forgive my younger self. Thanks a lot for the explanation! – The Oddler Dec 07 '20 at 17:16