Tagged Questions

Questions about real numbers not expressible as the quotient of two integers. For questions on determining whether a number is irrational, use the (rationality-testing) tag instead.

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Is there any attempt to explain irrational numbers from a geometrical point of view?

I'm trying to understand irrational numbers as the result of comparing different referential symmetries, and I'd like to know if there have been any attempt to explain irrationality from any ...
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Is there a fixed integer $n$ for which ${\pi}^{n}$ is prime number?

I would like to know the relationship between $\pi$ and prime numbers distribution ,then I would like to ask if there is a fixed integer for which ${\pi}^{n}$ can be prime or how do i disproof that ...
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Continuity - approximating an irrational number via rationals [duplicate]

If $x=p/q$, where $(p,q)=1$ are integers, then $f(x)=1/q$. If x is irrational then f(x)=0. Prove that: a) f is continuous for all irrationals b) f is not continuous for all rationals. I think ...
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What exactly are those “two irrational numbers” $x$ and $y$ such that $x^y$ is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
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Paper of Paul Erdös

I'm trying to understand On Arithmetical Properties of Lambert Series by Erdös, but am stuck on the first page. He states: Put $k=\left[(\log n)^{1/10}\right]$ and let $p_1,p_2,\ldots$ be the ...
Show $e^x$ is irrational for rational $x \neq 0$ [duplicate]
I want to show that if $x$ is rational and nonzero then $e^x$ is irrational. Clearly $e^{\frac{r}{s}} = \frac{p}{q} \Rightarrow q^s e^r = p^s$, but this doesn't seem helpful. The usual proof that $e$ ...