# Questions tagged [irrational-numbers]

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 the sum of partial sums $\frac{2\pi}{k}-{k \mod 2\pi}$ bounded?

Suppose, we are given sequence $a_k= k\mod 2\pi$ and sequence $y_{k,n}$, where $y_{k,n},0\leq n\leq k$ is just sequence $x_0,\dots,x_k$ sorted in increasing order for a given integer $k$. What can we ...
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### Do the decimal digits of $\pi$ and $\sqrt{2}$ coincide infinitely often?

I suspect the answer to the title question is "yes," but can it be proved? One would expect digits match in one tenth of cases.
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### Show that the following theorem is true or false: If $x\notin\Bbb Q$ then $x^2\notin\Bbb Q$. [closed]

In my approach, I attempted a proof by contrapositive, where it is assumed that $X^2$ is rational and $X$ is also rational. Therefore, we can represent $X^2$ with a ratio of two integers, $a$ and $b$...
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### Distribution of $\{n^p\alpha\}$ for irrational $\alpha$

Let $\alpha$ be an irational number. Consider sequence $x_n=\{n^p\alpha\}$, $n\in\mathbb{N}$ (it's the fractional part of $n^p\alpha$), where $p$ is a nonzero real number. Question. For which values ...
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### How to prove the irrationality of this number.

Prove that $\cos\frac{1}{\sqrt{2019}}$ is irrational. Would someone give me a hand on this question? Even a hint is appreciated. I don't have any clue.
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### Prove that this number is irrational it's involve the exponential [duplicate]

Prove that $2^{\sqrt{3}}$ is irrational or even it is transcendental? I am wondering if there exists the method that describes its property and its irrationality of these kinds (or even with some ...
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### Rational or Irrational?

Here is my question: can an Irrational number; like $e$, be equal to a second irrational number, $\pi$, times an integer, then divided by a second rational number? Such as: $e = \frac{\pi a }{b}$ , ...
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### Irrationality of the real number based on non-zero bits in the binary representations of primes

Assuming that $n$ is an odd natural number greater than $2$, let $B(n)$ denote a sequence of bits between the leftmost and rightmost non-zero bits of the base-$2$ representation of $n$. The ...
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### Why the first $x$ decimal places of $(5+\sqrt{26})^{x}$ are following a pattern?

$\sqrt{26}$ is irrational number, so the decimal places should show no pattern. But $(5+\sqrt{26})^{x}$ has these values: ...
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### $\pi$ is defined as the ratio of circumference to diameter, so why is it called “irrational”? [closed]

By definition, $\pi$ is the ratio of circumference and diameter of a circle. So, by definition itself it is a rational number. Why it is called irrational?
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### Copeland-Erdős constant: does the cumulative even digit count ever reach or overtake the cumulative odd digit count, and if so, at what prime?

Arthur Herbert Copeland and Paul Erdős proved in 1946 that the Copeland-Erdős constant is a normal number. Since all prime numbers other than 2 are odd, all prime numbers other than 2 end in an odd ...
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### Proving $\pi$ is irrational without arguing by contradiction

I know few proofs to prove that $\pi$ is irrational, but in all proofs one thing is common: it is done by the method of contradiction. So, I am looking for a proof without using contradiction. Is ...
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### Polynomial with natural coefficients at irrational points

I thought a lot at this problem: a polynomial $p(x) = a_{n}x^n + a_{n - 1}x^{n - 1} + ... + a_1x + a_0$, where $\forall i : a_{i} > 0$ (so that means that $p(x)$ includes all non-negative integer ...
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### Is it possible to convert an irrational number into rational by using a different number system?

It is certainly possible to convert a repeating number into a non repeating number by changing its number system. For example, 1÷3 is a repeating number in decimal as it equates to 0.3333333333......
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### “If $a+a^3$ is irrational, then $a+a^2$ is also irrational”.

Given $a\in\mathbb{R}$, define $p=a+a^2$ and $q=a+a^3$. Show that the following statement is false: "If $q$ is irrational, then $p$ is also irrational". My approach: I just was lucky to find a ...
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### Multiplication of two irrational to give rational

I know that we can obtain any rational $r\in\mathbb{R}\setminus\{0\}$ by the multiplication of two irrational numbers. There are many beautiful answers to this here. But I want to know that if there ...
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### Is it possible to have $\text{transcendental}^\text{transcendental} =\text{algebraic}$?
We know that it possible to have $\text{irrational}^\text{irrational}=\text{rational}$. To verify the possibility of this, we consider the expression $\sqrt{2}^\sqrt{2}$, if it is rational, then this ...