Articles of irrational numbers

a square root of an irrational number

I wonder if a square root of an irrational number is always irrational? I would tend to think that yes, but I can´t think of any justification. Also there are cases which are rather hard to decide like sqrt(Pi).

Asymptotic behavior of $\sum_{j=1}^n \cos^p(\pi u j)$ for large $n$ and $p$?

Consider the sum $$S=\sum_{j=1}^n \cos^p(\pi u j),$$ where $n$ and $p$ are positive integers and $u$ is irrational. Let’s say $p$ is even. I’m interested in the asymptotic behavior of this for $n$ and $p$ both large. This is my attempt to make a finite sum similar to the series in this problem that might […]

Density of positive multiples of an irrational number

Let $x$ be irrational. Use $\{r\}$ to denote the fractional part of $r$: $\{r\} = r – \lfloor r \rfloor$. I know how to prove that the following set is dense in $[0,1]$: $$\{\{nx\} : n \in \mathbb{Z}\}.$$ But what about $$\{\{nx\} : n \in \mathbb{N}\}?$$ Any proof that I’ve seen of the first one […]

How would you prove that $\sqrt{2}$ is irrational?

This question already has an answer here: Is $n^{th}$ root of $2$ an irrational number? [duplicate] 2 answers

Denseness of the set $\{ m+n\alpha : m\in\mathbb{N},n\in\mathbb{Z}\}$ with $\alpha$ irrational

This question already has an answer here: For every irrational $\alpha$, the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$ [closed] 3 answers

Does $\sin n$ have a maximum value for natural number $n$?

In formal, does there exist $k\in\mathbb{N}$ such that $\sin n\leq\sin k$ for all $n\in\mathbb{N}$?

Can permutating the digits of an irrational/transcendental number give any other such number?

Let $x_n$ be the infinite sequence of decimal digits of a fixed irrational/trascendental number. Can I obtain any other irrational/trascendental number’s sequence of decimal digits through a permutation of $x_n$?

Is the constant $e$ infinitely long?

The number $e = 2.718281828…$ is the base of the natural logarithm. Its decimal representation is infinitely long. Why does this mathematical constant contain an infinite number? What is the reason behind this? added for clearance: it contains infinitely long numbers, which does not repeat itself, how is this proven? it should at some point […]

Is there any sans-calculus proof of irrationality of $\pi$?

Is there a proof that will convince someone who doesn’t understand calculus, of $\pi$’s irrationality .

How to know that irrational numbers never repeat?

How would you respond to a middle school student that says: “How do they know that irrational numbers NEVER repeat? I mean, there are only 10 possible digits, so they must eventually start repeating. And, how do they know that numbers like $\pi$ and $\sqrt2$ are irrational because they can’t check an infinite number digits […]