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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4
votes
4answers
324 views

Does every irrational number contain arbitrarily long sequences of some digit?

Suppose we have an irrational number represented in base 3 such that all sequences of 1's and 2's in its ternary expansion have length less than some $n $. Does this imply there are arbitrarily long ...
8
votes
1answer
87 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 ...
9
votes
4answers
3k views

When is $\sin(x)$ rational?

Obviously, there are some points (e.g. $\pi$, $30^\circ$) but I am unsure if there are more. How can it be proved that there are no more points, or, if there are, what those points will be? EDIT: I ...
4
votes
2answers
35 views

Is $e^{\sqrt{2}}\gt 3$ or $e^{\sqrt{2}}\lt 3$

$e^{\sqrt{2}}\gt 3$ or $e^{\sqrt{2}}\lt 3$ which one holds true $?$ I know that $2\lt e \lt 3$ and $\sqrt{2}\gt 1$. Little help on how to use them to find the right inequality. ...
-3
votes
2answers
64 views

Prove or disprove that the sum of two irrational numbers is irrational [duplicate]

Prove or disprove that the sum of two irrational numbers is irrational. How do i answer this? Thanks.
1
vote
2answers
43 views

Intuitive explanation of the Dirichlet function and rationality

The Dirichlet function is defined by $f(x)=\begin{cases} c &\text{ if } x\in \mathbb{Q}\\d &\text{ if } x\notin \mathbb{Q}.\end{cases}, c\neq d$ See MathWorld's page for the full definition. ...
0
votes
0answers
49 views

Complexity of irrational numbers [duplicate]

What is the digit at $100^{\mathrm{th}}$ place after the decimal in the number $\left( \sqrt{2} + 1 \right)^{3000}$?
2
votes
0answers
22 views

Smallest number of workers in factory, Diophantine approximation

Q. In a factory, the percentage of male workers was $53.7802\%$ (rounding to nearest fourth decimal place) last year. What is the smallest number of female workers working there? Hint: Diophantine ...
5
votes
2answers
73 views

Closeness of $n! \ x$ to integers for irrational $x$

This question came up in the comments to another question. Is there an irrational number $x$ such that, for sufficiently large $n$, the product $$ n! \ x $$ is arbitrarily close to an integer? More ...
6
votes
12answers
4k views

Are there any irrational numbers that have a difference of a rational number?

Are there any irrational numbers that have a difference of a rational number? For example, if you take $\pi - e$, it looks like it will be irrational ($0.423310\ldots$) - however, are there any ...
3
votes
2answers
128 views

Slick proof that if an open set contains $\mathbb Q$ it has all irrational numbers, except a countable amount.

Basically I need help in proving that if $U\supseteq \mathbb Q $ is an open set in $\mathbb R$ with the usual topology then $\mathbb R \setminus U$ is countable. I'm not really sure how to proceed. ...
1
vote
4answers
636 views

If $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational [closed]

Prove that if $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational. I can understand how it works in my head, I don't know how to prove it though.
2
votes
0answers
17 views

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 ...
2
votes
1answer
36 views

Prove that the quotient of a nonzero rational number and an irrational number is irrational

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. ...
0
votes
1answer
55 views

Proving $\pi$ irrational: help with Lambert's proof. “Circularity”?

This expression is irrational. $$\tan(x)=\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-...}}}$$ But then he used the fact that $\tan{\frac{\pi}{4}}=1$, so $\frac{\pi}4$ is irrational. But how can we use ...
-1
votes
1answer
147 views

Is the sum of all irrational numbers between two integers rational?

Is the sum of all irrational numbers between any two integer constants rational? I think it should be, because every irrational number should have another irrational with which it would sum to a ...
7
votes
7answers
4k views

How to show that the product of two irrational numbers may be irrational?

Show that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers. All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}\cdot ...
27
votes
4answers
628 views

Linear independence of the numbers $\{1,e,e^2,e^3\}$

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$? The proof should not use that $e$ is transcendental. $e:$ Euler's number. $\{1,e,e^2\}$ is linearly ...
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?
0
votes
5answers
186 views

best approximation of $\sqrt{2}$

The approximation \begin{align} \sqrt{2} &\approx \frac{1}{8} \operatorname{csch}\left(\frac{3\pi}{2}\right) \operatorname{sech}^3(\pi) \, \left[2+3 \, ...
4
votes
2answers
109 views

First 10 digits after decimal point in the number $(1+\sqrt{3})^{2015}$

The question is how to find first 10 digits after decimal point in the number $(1+\sqrt{3})^{2015}$. I keep running into this kind of problems in a context of symmetric polynomials.
4
votes
1answer
56 views

$\lfloor x^k \rfloor \equiv m \pmod{n}$ with $x$ irrational

Let $x>1$ be an irrational number, and $n$ a positive integer. Is it true that, for each integer $m$, there exists an integer $k$ such that $$ \lfloor x^k \rfloor \equiv m \pmod{n}? $$
6
votes
3answers
228 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 ...
-2
votes
7answers
176 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 ...
0
votes
2answers
39 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.
0
votes
2answers
82 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
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
1answer
362 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
2
votes
3answers
5k views

Proving/Disproving Product of two irrational number is irrational

I saw this question where I had to prove/disprove that: Ques. Product of two irrational number is irrational. I tried 'Proof by Contraposition'. Product of two irrational number is irrational. p ...
1
vote
2answers
63 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 ...
9
votes
1answer
279 views

If $m\geq2$ is an integer, then $\sum\limits_{n=1}^{\infty}m^{-n^2}$ is irrational

Let $m \geq2$ be an integer. I want to ask how to prove that the sum of the following series is irrational: $$\sum _{n=1}^{\infty} \frac{1}{m^{n^2}}$$
1
vote
2answers
80 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
73 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
42 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?
53
votes
15answers
2k 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 ...
4
votes
1answer
63 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
70 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
51 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 ...
1
vote
2answers
180 views

Does there exist a function that is continuous at every rational point and discontinuous at every irrational point? And vice versa?

Actually there are 2 questions, but they are closely related. Does it exist a function that is: Continuous at every rational point and discontinuous at every irrational point? Continuous at ...
-6
votes
2answers
67 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.
16
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
43 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
76 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.
13
votes
1answer
115 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
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 ...
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
38 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
278 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
8
votes
1answer
126 views

Number made from ending digits of primes

Consider the number $0.23571379391713739171393971379371799173739113791379391173917133713717793$ ... The number is formed by the ending digits of the prime numbers. Is it known whether this number ...
1
vote
3answers
77 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 ...