# 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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### Irrational numbers generated by a deterministic cellular automaton?

If we consider a simple 1D cellular automaton (acting on a binary string) and record a value at a fixed position in the string, we can interpret the recorded sequence as a binary number. Most simple ...
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### show that this statement is false (counterexample) if $a,b \in \mathbb R \backslash \mathbb Q$ then $a \cdot b \in \mathbb R \backslash \mathbb Q$

if $a,b \in \mathbb R \backslash \mathbb Q$ then $a \cdot b \in \mathbb R \backslash \mathbb Q$ Okay so the question asks to show, with a counter example, that the above statement is false. Here ...
363 views

### Proof by Contradiction relating to rational and irrational numbers

I've been given the question: given $x,y\in\mathbb{R}\setminus\mathbb{Q}$ and $x+y =\frac{m}{n}$, prove $x-y$ is irrational. I tried solving this using a proof by contradiction but I feel like I got a ...
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### Prove $\cos\frac{\pi}{2^{n+1}}$ is irrational

Prove that for every number $n\in\mathbb N$,number $\cos\frac{\pi}{2^{n+1}}$ is irrational. I really don't know where to start.
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### $\pi \not\in \mathbb{Q}$?

I've taken this fact for granted; some thinking tells me that indeed, I cannot express it with fractions. So it's not rational. But well, if $p,q \in \mathbb{Q}$ then $p+q \in \mathbb{Q}$ since it is ...
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### Rational numbers as angles - where do irrationals fit in?

If we make a rectangular grid with integer coordinates, it's possible to assign a unique angle to any rational number, using the definition $\tan \phi=y/x$ for $\phi \in (-\pi/2, \pi/2)$. For ...
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### How to make continued fractions of any number?

I recently found an continued fraction representation of $\pi$, and I wondered how can I make an continued fraction that converges into a number? The MAIN question is: how do you make a continued ...
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### Prove: $\tan\frac{\pi}{24}=2\sqrt{2+\sqrt{3}}-\sqrt{3}-2$

How to prove that $$\tan\frac{\pi}{24}=2\sqrt{2+\sqrt{3}}-\sqrt{3}-2$$ I get $$\tan\frac{\pi}{24}=\sqrt\frac{2\sqrt{2}-\sqrt{3}-1}{ 2\sqrt{2}+\sqrt{3}+1}$$ but i can't transform it.
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### When is a finite sum of powers of non-integer a rational number? [closed]

Concretely, is there $b \in \mathbb R, n,k \in \mathbb N$ such that $\sum_{i = n}^{n+k} b^i \in \mathbb Q$ ?
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### Bijection between $[0,1)$ and the space of binary sequences

My question deals with the problem of showing that the set $$\Omega = \{ \omega \colon \omega =(a_1,a_2, \ldots ), a_i =0,1\}$$ has the same cardinality as the interval $[0,1)$. In a textbook I read ...
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### Show that a certain number defined via its decimal expansion is not rational

For each function $f:\mathbb{N}\to \mathbb{N}$ we define the real number, in decimal notation $A(f)=0.f(0)f(1)f(2)f(3)\ldots$. Show that, if $f(x) =x^2$, then $A(f)=.0149162536\ldots$ is irrational....
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### Cubic polynomial with three (distinct) irrational roots

I am looking for an equation $$x^3+ax^2+bx+c=0, \qquad a, b, c \in \Bbb Z,$$ of degree $3$ that has $3$ different roots. For an equation of degree $2$ it is easy---for example $x^2-2=0$---but I ...
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### Rational or Irrational number [closed]

we know that "$a$" is a Irrational number .But "$a^2+a$" is Rational. Can You find "$a$"? (more than one answer is available)
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### For what numbers $n$ is $\sqrt{n}$ irrational?

I would say it has something to do with the numbers that can be expressed as a factor of different prime numbers, but when I get to $8$, that can be changed to $2^3$, which goes against this. Is there ...
### Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?
Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$? I know that \begin{align*} \mathbb{Q}(\sqrt{2}) &= \{a+b\sqrt{2} \mid a,b \in \mathbb{Q}\}, \\ \mathbb{Q}(i) &= \{a+...