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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5
votes
1answer
93 views

When $\cos(\theta) = 1/8$ it's easy to show $\theta$ is an irrational angle. Is it algebraic?

Along the lines of my lines of my previous question about irrational angles "$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?", I was working on ...
6
votes
3answers
213 views

Why is the remainder uniformly distributed when 1,2,3,… are divided by an irrational number?

Let remainder $r$ be defined as $$ r = n - pq $$ where $n \in \mathbb{N}$ is the dividend , $q \in \mathbb{R}$ is the divisor, and $p = \mathrm{floor}(n/q)$. I calculated the remainders by dividing ...
0
votes
2answers
37 views

Show that an irrationally periodic function is also a constant function [duplicate]

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$, and any real number $x$ we have $f(x)=f(x+r)$. Show that $f$ is a constant function.
1
vote
1answer
31 views

Imprecise logarithms that reference sets of numbers.

I apologize in advance if my question seems vague, I'm only in algebra II, so It may turn out that I lack the terminology to phrase my question correctly. Some background, we just finished our unit ...
-2
votes
7answers
94 views

$0.333333$ - a recurring or non-terminating decimal?

I have read like, 1.All terminating and recurring decimals are RATIONAL NUMBERS. 2.All non-terminating and non recurring decimals are IRRATIONAL NUMBERS. if the statements are right, then here ...
2
votes
3answers
85 views

Explain the proof that the root of a prime number is an irrational number

Though the proof of this is done in a previous question i have a doubt in a certain concept so i ask to clear it.In the proof we say that $\sqrt{p} = \frac{a}{b}$. (In their lowest form.) $p = a^2 ...
1
vote
2answers
59 views

How to check this number $\sqrt{47}$ is irrational [duplicate]

Prove that $\sqrt{47}$ is irrational number. I know that a rational number is written as $\frac{p}{q}$ where $p$ & $q$ are co-prime numbers. But I do not have any idea to prove it irrational ...
2
votes
2answers
55 views

If $a \in \mathbb{I}$ , how is $\overline{\mathbb{Z}+ a\mathbb{Z}}=\mathbb{R}$

If $a \in \mathbb{I}$ , how is $$\overline{\mathbb{Z}+ a\mathbb{Z}}=\mathbb{R}$$ It says in my notebook that this set in dense in $\mathbb{R}.$ How do I prove this density? With say $\mathbb{Q}$ and ...
3
votes
1answer
68 views

Sum of square root of non perfect square positive integers is always irrational?

Let $S$ be a set of positive integers such that no element of $S$ is a perfect square. Is it true that $\sum_{s_i \in S} \sqrt{s_i}$ is always irrational? Motivation. Suppose the length of the ...
2
votes
2answers
35 views

Can a non-rational polynomial be rational at all integers?

Is there a polynomial $f \in \mathbb{R}[X]$ such that for every $x \in \mathbb Z,\>\> f(x)$ is rational but at least one of the coefficients of $f$ is irrational?
4
votes
1answer
60 views

Approximation of irrational numbers?

Problem Suppose $\theta>1$ is an irrational algebraic integer, i.e. $\theta\not\in\mathbb Z$ but satisfies a monic polynomial with integer coefficients, and $\{a_n\}_{n\ge0}$ is a sequence of ...
2
votes
0answers
67 views

What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
0
votes
0answers
44 views

Proving transcendental numbers

I'll warn now that this is probably a big question, but I am wondering if anyone can explain why it is so difficult to prove whether a number is transcendental or algebraic. For example, it is now ...
-6
votes
2answers
65 views

Is a number of the form $p+p^2$ be ever rational? [closed]

Is $p+p^2$ ever rational when $p$ is an irrational number? Also, if not please provide a proof.
15
votes
4answers
1k views

Are there more transcendental numbers or irrational numbers that are not transcendental?

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it ...
-4
votes
4answers
39 views

Is the cubed root of x irrational if and only if x is irrational?

Is the cubed root of x irrational if and only if x is irrational? Hoping for simple answers. Thank you very much.
0
votes
1answer
75 views

Are $x,y$ rational if $x+y$ is rational and $x-y$ is rational? [closed]

Are $x,y$ rational if $x+y$ is rational and $x-y$ is rational? This question was given in maths class, and I don't know where to start. I would be happy if the answer was included in the proof.
0
votes
1answer
32 views

The probability of a number appearing in an approximation of an irrational number?

I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 ...
13
votes
1answer
106 views

$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?

I've been working on a problem related to the 3x3x3 Rubik's Cube where you allow faces to be turned by $45^\circ$ instead of just the usual $90^\circ$. We know for the standard 3x3x3 the cube is ...
0
votes
1answer
20 views

Convergence and Irrationality of $\frac{H_{(n,-n)}}{(n+1)^n}$ as $n$ approaches infinity

We define $H_{(a,b)}$ as the $a^{th}$ harmonic number of class $b$. In other words, $$H_{(a,b)}=\sum_{k=1}^a \frac{1}{k^b}$$ More information about generalized harmonic numbers can be found here. Let ...
1
vote
1answer
37 views

Limit of a function - $x$ either rational or irrational - limit $1$ or $0$. [duplicate]

Show that: The continuous functions $f_{n,k}(x):=(\cos(k!\pi x))^{2n},0\leq x \leq 1$ satisfy the relation $\lim_{k\to \infty}(\lim_{n\to \infty}f_{n,k}(x))=\begin{cases} 1, & \textit{if ...
3
votes
5answers
267 views

Is :$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$ irrational or transcendental or real number?

Is there someone who can show me if :$$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$$ is irrational or real or transcendental number ? Thank you for any help
1
vote
3answers
74 views

Some questions about proofs of irrational numbers

I have some questions about some things I want to clarify in regard to basic questions that ask to show that roots are irrational, for example $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{6}$. To me, I think ...
1
vote
1answer
46 views

Randomness in pi and other irrational numbers [duplicate]

This is a post I read about pi while looking for stuff about tau -which is two times as much as pi. This makes me wonder, why does only pi contain such randomness? Don't other non-repeating and ...
0
votes
1answer
61 views

About the vertices of a regular polygon in the plane having rational coordinates [closed]

I have to prove that, except in the case $n=4$, the vertices of a regular $n$-agon in the Euclidean plane cannot have all rational coordinates $(x,y)$. Some idea?
2
votes
1answer
52 views

Does a bijection from the reals to the any binary form?

It is fairly simple to store all rational numbers in a binary format (not base 2) (a language composed of only 1s and 0s, no . marking) by simply storing one integer, a seperator, and another integer. ...
-1
votes
1answer
77 views

If $\pi$ is not algebraic number then : is $\pi ^{n}$ algebraic number for $n >1$?

if $\pi$ is not algebraic number then : is $$\pi ^{n}$$ algebraic number for $n >1$ ? Thank you for any kind of help .
5
votes
2answers
80 views

LCM of irrationals

So, I was recently asked by a friend about the lcm of two irrational numbers. As far as I know, mathematically speaking, lcm is generally defined only for positive integers (and sometimes extended to ...
3
votes
1answer
266 views

Is a non-zero integral multiple of an irrational number guaranteed to be irrational?

I got to wondering if a non-zero integral multiple of any irrational number is guaranteed to be irrational? This seems intuitive but I can't prove it to myself. There is an answer on this site with ...
2
votes
0answers
60 views

Conway's box function generalized as a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
0
votes
1answer
38 views

Riemann integral of a non continuous function

We have a function $f : I=[0,1] \rightarrow \mathbb{R}$ defined as: $$f(x)=\begin{cases} 1 &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$ a) Show ...
3
votes
0answers
58 views

Transcendence of $\Gamma(1/3), \Gamma(1/4)$

Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a reference to that proof? Or maybe some details on the essential ideas ...
1
vote
0answers
198 views

Very simple proof that $\sqrt{2}$ is irrational.

I came across a nice-looking proof that $\sqrt{2}$ is irrational here. It somehow seems to good to be true. What are the assumptions being made in the proof and if this proof is indeed correct, why is ...
3
votes
0answers
40 views

Do the second-last-digits of the primes $\ge 11$ form a transcendental number?

Suppose, the number $x$ is constructed from the second-last-digits from the primes $\ge 11$ The first $1996$ digits of $x\ =\ 0.11112...$ after the decimal point are : ...
1
vote
1answer
44 views

The irrationality of rapidly converging series

I recently saw a pretty elegant proof of the irrationality of $e$, namely: Let $s_n:=\sum_{k=0}^{n}{\frac{1}{k!}}$ such that $e=\lim_{n\to\infty} s_n$. We obviously have $s_n<e$ and furthermore ...
3
votes
1answer
71 views

Unknown Possible Irrational Number Determination

I have a particular sequence, the coefficients determined by the generating function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty\varepsilon_n\frac{x^n}{n!}$$ The first few numbers are ...
-4
votes
1answer
42 views

The product of xy of two real numbers x and y is irrational then at least one of the x or y must be irrational. [duplicate]

Prove if true or find a counterexample.... The product of $x y $ of two real numbers $x$ and $y$ is irrational then at least one of the $x$ or $y$ must be irrational.
0
votes
1answer
64 views

why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?

When I was comparing proof for $\sqrt{2}$ and sum of two rational numbers, I found that the proof of two rational number did not mention anything about common factor in the ratio. one proof I found ...
1
vote
5answers
76 views

Why the set of irrational numbers is represented as $\mathbb{R}\setminus\mathbb{Q}$ instead of $\mathbb{R}-\mathbb{Q}$?

What does the "\" symbol means in this context? I have seen it used for quotient sets like $X /{\sim}$ where $X$ is a set and $\sim$ is an equivalence relation but I don't know what it means applied ...
51
votes
15answers
1k views

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
0
votes
2answers
62 views

Finding $n$th root of 2 is irrational using given polynomial

The polynomial $f(x)$ is defined by $f(x)=x^n + a_{n-1}x^{n-1}+ \cdots + a_{2}x^2+a_1x+a_0$ where $n \geq 2$ and the coefficients $a_0, \cdots, a_{n-1}$ are integers, with $a_0 \neq 0$. ...
0
votes
0answers
30 views

On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post. I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / ...
3
votes
2answers
95 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
28
votes
12answers
2k views

Computing irrational numbers

I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?
16
votes
4answers
238 views

Homework 8th grader: $\pi^2$ is irrational

I'm tutoring a girl in 8th grade (so she is 14 years old) and she recently had a mathematics chapter about numbers. In the last paragraph they introduced the difference between rational and irrational ...
1
vote
2answers
40 views

Show: t(x) = x√2 + √3 is irrational. Hint: consider t(x)²

Earlier in the question we were asked to show that the square root of 6 is irrational, which I did. But I can't seem to figure the last part out. I have included an image for reference. Help is ...
8
votes
0answers
75 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
-1
votes
2answers
44 views

Does an irrational number $C$ exist such that $C \cdot \sqrt 2 \in \Bbb{Q}$?

Does an irrational number $C$ exist such that $C \cdot \sqrt 2 \in \Bbb{Q}$, where $\sqrt2 \not\mid C$? I just thought of this, I'm trying to find answers that aren't of the form $C=a\sqrt2, ...
44
votes
9answers
8k views

Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
4
votes
1answer
25 views

Several values of irrational exponentiation

When talking about a number to a rational exponent, there are as many answers as the denominator of the exponent. Like the question: Is $9^{1/2}$ equal to $3$ or $-3$. However when we have an ...