Infinite number of irrational numbers between 0 and 1?

Question

Basically, I am asking if this is true.

$$|\{z\mid 0<z<1\land z\in\mathbb{R-Q}\}|=\infty$$

But I don't know how to prove it.

Answer 1

$1/\sqrt{n}$ is irrational for every prime number $n$. I'm leaving uncountably many out but this proves that there are infinitely many.

Answer 2

It's true, and there are many many ways to prove it.

Taking any rational number $q$ such that $0<q<\pi$, the number $\frac{q}{\pi}$ is an irrational number between $0$ and $1$, and since there are infinitely many rationals between $0$ and $\pi$, there must be infinitely many irrationals between $0$ and $1$.

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Or, you could say that for every $n$, there exists an irrational number between $\frac{1}{n+1}$ and $\frac1n$.

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Or, you could go decimal. There are infinitely many non-repeating strings of digits from $0$ to $9$

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Or, take any irrational number $x$ on $(0,1)$ (for example, $x=\frac{1}{\sqrt 2}$. Then, $x,\frac x2,\frac x3,\dots$ are all irrational and all on $(0,1)$.

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Alternatively, the set $(0,1)$ is uncountably infinite, while $(0,1)\cap \mathbb Q$ is countably infinite, so the set $(0,1)\cap\mathbb R-\mathbb Q$ must be uncountably infinite as well.

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