Is [0.1, 1[ countable? (Or, why can a decimal number have an infinite of digits, but a whole number cannot?)

Question

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?

Answer 1

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.

Answer 2

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 $[0.1,1]$.