# 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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### The irrationality of $\pi/e$ is listed as open yet the infinite product formula for it seems to suggest a way to prove it.

And the formula of all rational products seems to suggest that taking some n as n approaches infinity, the formula will have an always increasing amount of uncancelled primes(so provably non ...
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### Rationalising factor of $a+b \sqrt{2}+c \sqrt{3} + d \sqrt{6}$

I am trying to express the inverse of $a+b \sqrt{2}+c \sqrt{3} + d \sqrt{6}$ (given $a, b, c, d \in \mathbb{Q}$) in the form $e+f \sqrt{2}+g\sqrt{3}+h\sqrt{6}$ (where $e, f, g, h \in \mathbb{Q}$). I ...
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### How to prove $\log_23$ is irrational?

I think using contradiction is good. Assume $\log_23$ is rational Then $\exists p\in \Bbb{Z}, q\in \Bbb{Z}^*: \log_23 = \frac{p}{q}$ ###$p, q$ has no common factors. Then $3^{q}=2^{p}$ ... Here ...
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### If $x$ is rational and $xy$ is irrational, then $y$ is irrational. [closed]

This is a statement that I need to prove. Let $x$ and $y$ be real numbers. If $x$ is rational and $x\times y$ is irrational, then $y$ is irrational. I believe you have to prove this using ...
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### Suppose that a sequence of rational fractions p/q converge to an irrational number

Suppose that a sequence of (rational) fractions p/q converge to an irrational number r. Show that q converges to infinity.
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### Rational Points on $\sin x$ and $\cos x$

Are there any values for $x$ such that both $\sin x$ and $\cos x$ are rational besides $\displaystyle\frac{n\pi}{2}$ and $n\pi$, where $n$ is an integer? I also only want to include $x$ values that ...
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### Is the ratio of two natural logarithms irrational or rational?

Is there any way to prove that the ratio of two natural logarithms is rational or irrational? Take the natural logarithms of $a = 25$ and $b = 6$, for example. Can you prove $\ln(a)/\ln(b)$ rational ...
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### Are $\frac{\pi}{e}$ or $\frac{e}{\pi}$ irrational?

Is it clear whether $\displaystyle \frac{\pi}{e}$ or $\displaystyle \frac{e}{\pi}$ are irrational or not? If not, then there would exist $q,p\in \mathbb{Z}$ such that $$p\cdot \pi = q\cdot e$$
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### Is there an irrational number arbitrarily close to another irrational number?

I know that there is a rational number arbitrarily close to an irrational, due to the density of real number. But what about an irrational number? Thanks!
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### if $|f(x)-f(y)|\le |x-y|^{\sqrt 2}$ then is $f$ a constant function?

if $f: \mathbb{R}\to \mathbb{R}$ satisfies $$|f(x)-f(y)|\le |x-y|^{\sqrt{2}}$$ for all $x,y\in \mathbb{R}$ ,then is f increasing ,decreasing or constant? in my view ,it is clear that $|f(x)-f(y)|$ is ...
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### Prove that $\sqrt{6}-\sqrt{2}$ $> 1$.

I'm trying to prove that $\sqrt{6}-\sqrt{2}$ $> 1$. I need to admit that I'm completely new to proof writing and I have completely no experience in answering that kind of questions. However, I came ...
### Let a,b be rationals and x irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$.
I'm trying to solve the following problems: Let $a$,$b$ be rationals and $x$ irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$ Let $x$,$y$ be rationals such that \$\frac{x^2+x+\sqrt{...