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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7
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2answers
134 views

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 ...
1
vote
2answers
60 views

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 ...
6
votes
3answers
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 ...
1
vote
1answer
72 views

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.
1
vote
4answers
78 views

$\pi \not\in \mathbb{Q}$?

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 ...
3
votes
0answers
45 views

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 ...
1
vote
0answers
37 views

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 ...
0
votes
2answers
66 views

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.
0
votes
2answers
46 views

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$ ?
3
votes
3answers
56 views

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 ...
1
vote
1answer
26 views

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....
1
vote
4answers
90 views

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 ...
2
votes
3answers
209 views

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)
0
votes
1answer
31 views

Pythagorean Theorem on Spiral of Theodorus Triangles

I have 1 right triangle of dimensions $\sqrt75$$, 11, 14$. I'd like to know how to quickly obtain the other right triangles with $\sqrt75$ as a leg, and two integers as the hypotenuse and the other ...
0
votes
2answers
43 views

Irrational Numbers and their squares

If $s$ is irrational is $s^2$ irrational? Looking at example (a) $s= \sqrt 2$ then $s^2= 2$, which is rational but looking at example (b) $s= 5^{1/3}$, then $s^2= 5^{2/3}$ which is irrational or $\...
0
votes
2answers
44 views

Irrationality of $ 1/a + 1/b$

I have thought about this and was wondering if anyone could provide an example of real numbers $a$ and $b$ such that $a + b$ is rational but $1/a + 1/b$ is irrational or prove the statement false.
2
votes
2answers
42 views

Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational?

Suppose we wish to prove that the square root of two is irrational. We begin by assuming that it is rational. Namely, where both $a$ and $b$ are integers $$\frac{a}{b} = \sqrt 2 % MathType!MTEF!2!1!+...
5
votes
6answers
142 views

Proving that $2\sqrt 3+3\sqrt[3] 2-1$ is irrational

Prove that $2\sqrt 3+3\sqrt[3] 2-1$ is irrational My attempt: $$k=2\sqrt 3+3\sqrt[3] 2-1$$ Suppose $k\in \mathbb Q$, then $k-1\in \mathbb Q$. $$2\sqrt 3+3\sqrt[3] 2=p/q$$ I'm stuck here and don'...
0
votes
1answer
39 views

Division of Square Root of Primes are Irrational

Prove that for any distinct primes $p$ and $q$, the ratio $\frac{\sqrt p}{\sqrt q}$ is irrational. I know that separately $\sqrt p$ and $\sqrt q$ are irrational, so my initial thought process was to ...
10
votes
2answers
1k views

Reversing the digits of an infinite decimal

Let $x$ be a real number in $[0,1)$, with decimal expansion $$ x = 0.d_1 d_2 d_3 \cdots d_i \cdots \;. $$ If the decimal expansion is finite, ending at $d_i$, then extend with zeros: $d_k = 0$ for all ...
4
votes
6answers
227 views

How can never ending decimal numbers represent finite lengths? e.g. pi(π), $\sqrt{2}$

Recently, I was in a discussion with a colleague that, whether the πd really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came ...
9
votes
0answers
121 views

Infinitely nested radical expansions for real numbers

Conjecture. For any real number $x \in (0,1]$ there exists a unique expansion in the form $x=-2+\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}$ with $a_k$ being natural numbers from the set $(2,3,4,5,6)$. ...
1
vote
3answers
57 views

Rationalize a surd $\frac{1}{1+\sqrt{2}-\sqrt{3}}$

How can I rationalize the following surd $$\frac{1}{1+\sqrt{2}-\sqrt{3}}$$ What would be the conjugate of the denominator
1
vote
3answers
39 views

exhibit a countable set of irrational numbers with justification

So I have been given this problem and I am totally stumped on what to do...everything I have learned says the irrational numbers are uncountable but I am supposed to exhibit a countable set of ...
1
vote
1answer
42 views

Rationality of $a^2+b^2$

I have looked into this topic lately and have not found an answer to the following question. Is the following true: If $a,b\in\mathbb{R}$ and $a + b$ is rational, then $a^2 + b^2$ is rational
0
votes
1answer
62 views

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 ...
4
votes
2answers
62 views

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 ...
4
votes
1answer
84 views

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 ...
-1
votes
3answers
63 views

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 ...
-1
votes
2answers
67 views

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.
3
votes
2answers
69 views

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 ...
1
vote
1answer
37 views

Continous function from $ \Bbb Q \rightarrow \Bbb R $, $ f = 1 $ for $x > \sqrt2$ and $ f = 0$ for $x < \sqrt2$

I'm not really sure how to go about this problem. Show that $h : \Bbb Q \rightarrow \Bbb R $, with $$ h(x)=\begin{cases} 0 &\text{for $|x|< \sqrt{2}$} \\ 1 &\text{for $|x|>\sqrt{2}$} \...
1
vote
1answer
88 views

Which set is more dense: set of irrational numbers or set of rational numbers? [duplicate]

Is the infinity of irrational numbers equal to the infinity of rational numbers? Or is one is greater than other? And what is the proof? I could not find out a rigorous proof about this. P.S. I am ...
1
vote
1answer
70 views

Set of Rational numbers a countable set?

How can we say that rational numbers is a countable set? I can divide a rational number by infinite different number of natural numbers so shouldn't there be infinite rational numbers. http://www....
1
vote
0answers
38 views

Arithmetic geometric mean - irrational, algebraic, trancendental?

Are there some general theorems about rationality/irrationality and abgebraicity/transcedentality of arithmetic-geometric mean? At least for some group of numbers (like natural numbers)? Or even for ...
2
votes
2answers
88 views

The irrationality of $\sqrt[n]{2}$ from the FLT.

It's common to see the Fermat Last Theorem being used to prove the irrationality of $\sqrt[n]{2}$. In fact, according this post, the said proof appeared in American Mathematical Monthly. On the other ...
0
votes
2answers
66 views

Given a rational number and an irrational number, both greater than 0, prove that the product between them is irrational.

Does this proof I made make sense? Proof// $\mathbf a$ is the rational number, $\mathbf b$ is the irrational number. Assume that $\mathbf {a * b}$ is rational due to proof by contradiction. ...
1
vote
1answer
73 views

Geometric proof for irrationality of $\pi$

Is there a geometric proof for irrationality of $\pi$? That would be neat.
0
votes
0answers
36 views

How to prove $a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+…+a_{n}\sqrt[b_{n}]{c_{n}}$ is irrational?

Let's define the number $$A=a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+.....+a_{n}\sqrt[b_{n}]{c_{n}}$$ where $a_{1}, a_{2}, ..., a_{n}$ are positive integers and $b_{1}, b_{2}, ..., b_{n}, ...
3
votes
2answers
112 views

Elementary proof that $\pi$ is irrational

I'm trying to understand the first proof in this page. So we have $$S=\frac{\pi }{4}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}=S_{n}+R_{n}$$ where $S_{n}=\sum_{k=1}^{n}\frac{(-1)^{k-1}}{2k-1}$ and $...
9
votes
0answers
133 views

Swapping the digits of an algebraic number (e.g. $\sqrt 2$)

Let an algebraic number, say $ a=\sqrt 2 = 1.41421356237309504880...$, and define $$b=f(a)=1.14243165323790058408...$$ by swapping the digits $a_{2i+1}$ and $a_{2i+2}$ for $i≥0$, corresponding to ...
0
votes
1answer
26 views

Shift of finitely many reals into irrationals

This question came in one of class-room discussion. Given finitely many distinct real numbers $x_1,x_2,\cdots, x_n$, does there exists a real number $y$ such that $y+x_1, y+x_2, \cdots, y+x_n$ are ...
5
votes
5answers
133 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
0
votes
0answers
60 views

Why are positive rational numbers countable but real numbers are not? [duplicate]

If we can say that any positive rational number is countable or listable by showing that every positive rational number is the quotient of p/q of two positive ...
0
votes
1answer
14 views

Additivity Implies Homogeneity of Rational Scalars

I did my best to search for this question on the site- but I did not find it. Here it is: If a function $f:\mathbb{R}^2\to\mathbb{R}$ satisfies $f(u+v)=f(u)+f(v)$ for all $u,v\in\mathbb{R}^2$, ...
4
votes
1answer
101 views

Is $\arctan2$ irrational? [duplicate]

Is $\tan^{-1}2$ an irrational number or a rational number? How to show that? Or generally how to show $\tan^{-1}3, \tan^{-1}4, \tan^{-1}5...$ is irrational or rational?
2
votes
2answers
57 views

Square root of the product of consecutive natural numbers is irrational

Prove that for all $n\in\mathbb{N}$ the number $\sqrt{n(n+1)}$ is irrational. My first move would be: Let's assume that it's not, that it $\sqrt{n(n+1)} = \frac{a}{b}$, where $a,b\in\mathbb{N}$ and $...
3
votes
3answers
81 views

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 ...
5
votes
2answers
91 views

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+...
0
votes
1answer
35 views

$x^J = y$, $J = 2.455\ldots$ What's the rest of $J$?

I have a problem where I need to know what J is. I do x^J and get y. For example, if I do 5^J, I would want to get 55 as y. Same with 4^J = 30. When J is 2.455, it works up to 4 only! I need for ...