# Questions tagged [irrational-numbers]

An irrational number is a real number that cannot be expressed in the form $\frac{n}{m}$ where $n$ and $m$ are integers.

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### Could this weird number be rational, possibly equal to $(4\cdot 37 \cdot 337)/(15^{2}\cdot 17^{2})$?

I am desperately looking to find a simple sequence $(z_n)$ recursively defined, that leads to an irrational number $\lambda$ with the following properties: $z_n = x_n/q_n$ with $q_n$ a power of 2 and ...
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### Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$

Consider the series $$\sum_{n=1}^{\infty} ( \{ n \xi \} - \frac{1}{2})$$ where $\{ \}$ denotes the fractional part and $\xi$ is some irrational number. It clearly does not converge but can we show ...
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### Are mantissas of irrationals provably unique, at a given precision? [closed]

Many thanks to all responders! Is there any research as to the uniqueness of mantissas of irrationals? It's easy to see that the mantissa of the square root of 5 (0.236067977...) and the mantissa of ...
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### Irrationality or transcendence of $i^{i\Omega}$ and $2^\Omega$, with $\Omega=W(1)$ and $W(x)$ being the main branch of Lambert $W$ function

In this post we denote the main (or principal) branch of the Lambert $W$ function as $W(x)$, I add as reference that Wikipedia has the article Lambert $W$ function. The particular value $W(1)=\Omega$ ...
Let $a,b,c,d$ be positive natural numbers such that $\{a,b\} \neq \{c,d\}$ and such that none are perfect powers. Is it true that $$\frac{\log a \log b}{\log c \log d} \notin \mathbb{Q} ?$$ The ...