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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0
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2answers
84 views

Irrationality of $x$ if $x > 1$ and $x^x = 2$

Show that if $x>1$ and $x^x=2$, then $x$ must be irrational. I know you have to show that it cannot be reduced into a form $\frac pq$, but get stuck with quite ugly algebra.
4
votes
1answer
63 views

For integers $n \neq 0$ is $\sin n$ irrational or transcendental?

For integers $n \neq 0$ is $\sin n$ irrational or transcendental? This arose from another question. I would hypothesize yes and yes, possibly with proof for irrationality existing and but not for ...
1
vote
4answers
88 views

Is $|\sin(n)|\leq1$ or $|\sin(n)|<1$ for integer $n$?

$\pi$ is irrational, therefore there exist no finite integers $m,n$ such that $n=(m+\frac{1}{2})\pi$, therefore there is no $\sin(n)=\pm1$. So if n defined to be a finite integer, I am comfortable ...
0
votes
0answers
71 views

How do we know $\pi$ cannot be expressed a root [duplicate]

In other words, is there a proof that $\pi^a\neq b$ where $a,b\in \mathbb{Z}$?
0
votes
3answers
229 views

What are irrational real numbers?

I was given a question saying: "One can show that the union of two countable sets is countable. Is the set of irrational real numbers countable?" I don't know what irrational real numbers are....
8
votes
4answers
135 views

Is there a pythagorean triple such that all angles of the corresponding triangle are simple fractions of $\pi$?

Obviously, the most interesting pythagorean triple $(a, b, c)$ would be one for which the corresponding triangle (with integer side lengths $a, b, c$) has angles 90°, 60° and 30° ($\frac{\pi}{2}, \...
1
vote
1answer
65 views

Is the Champernowne constant actually useful

Has the aforementioned constant ever been used in any major proofs? Can it be expressed in terms of $e$, $\pi$, or both? Does it appear in any sort of geometric sense like $\pi$ does? Or is it just a ...
-5
votes
1answer
879 views

Why 1/3 is rational? [closed]

Could we say we know how it is going to behave so it is rational. But we do not know how pi is going to behave.
4
votes
1answer
57 views

Compare $\log_34$ and $2^\frac 1 4$

Both $\log_34$ and $2^\frac 1 4$ are somewhere in between $1$ and $2$. I know I can get approximate values but they are difficult to calculate by hand so I conclude it's not the way I'm supposed to do ...
0
votes
2answers
98 views

How to prove that $x=0.1234567891011\dots $ is irrational? [duplicate]

I'm in $9th$ class and I was wondering how to solve this problem. I only know how to prove that $0.1010010001\dots$ is irrattional.
0
votes
4answers
298 views

Random irrational number generator?

Is it possible to create a algorithm that will generate irrational numbers $0<x<1$ with a density that is uniform down a specified resolution? Would such an algorithm be necessarily limited to ...
31
votes
2answers
600 views

Show that $e^{\sqrt 2}$ is irrational

I'm trying to prove that $e^{\sqrt 2}$ is irrational. My approach: $$ e^{\sqrt 2}+e^{-\sqrt 2}=2\sum_{k=0}^{\infty}\frac{2^k}{(2k)!}=:2s $$ Define $s_n:=\sum_{k=0}^{n}\frac{2^k}{(2k)!}$, then: $$ s-...
2
votes
1answer
48 views

How to express $15.3\dot{9}$ in fractional form

In the number $15.3\dot{9}$, $9$ is repeated forever. If the number is rational then it can be expressed as a fraction (i suppose it is rational since it's an exercise for me to find it's rational ...
5
votes
2answers
43 views

Can we define sum and product of two irrational numbers using Cauchy sequences of their simple continued fraction convergents?

There is a lot of questions about sum and product of irrationals here, so I hope you'll bear with me. Simple continued fraction is a very convenient way to represent any number since every real ...
6
votes
1answer
158 views

What are some of the implications of $\pi + e$ being rational?

Whether or not $\pi + e$ is rational is an open question. If it were rational, what would some of the implications be?
6
votes
1answer
403 views

What's an example of a number that is neither rational nor irrational?

Of course in regular logic, the answer is there aren't any. But in intuitionistic logic, there might be, as seen by this answer: http://math.stackexchange.com/a/1437130/49592. My question is, as per ...
3
votes
1answer
153 views

Denseness of the set $\{ m+n\alpha : m\in\mathbb{N},n\in\mathbb{Z}\}$ with $\alpha$ irrational [duplicate]

How to prove that the set $\{ m+n\alpha : m\in\mathbb{N},n\in\mathbb{Z}\}$, ($\alpha$ is an irrational number) is dense in $\mathbb{R}$? Using the fact every additive subgroup of $\mathbb{R}$ is ...
1
vote
1answer
46 views

Prove that for any natural number $3\leq n$ and rational number t, such that $0<t<1$ , $(t^{n}+1)^{‎\frac{2}{n}}‎$ is irrational.

Prove that for any natural number $3\leq n$ and rational number t, such that $0<t<1$ , $(t^{n}+1)^{‎\frac{2}{n}}‎$ is irrational or provide a counterexample.
6
votes
1answer
189 views

Another conditionl leading to irrationality of $\sum _{k=1}^ \infty \dfrac 1{n_k}$?

If $\{n_k\}$ is a strictly increasing sequence of positive integers such that $\lim \inf _{k \to \infty} n_k ^{1/2^k} >1$ and $\lim _{k \to \infty} n_k^{1/2^k}$ does not exist , then is it true ...
4
votes
1answer
62 views

Limit points of particular sets of real numbers.

How to find limit points of the following sets of real numbers, for irrational $\alpha$ ? $(1)$ $\{ m+n\alpha:m,n\in\mathbb{Z}\}.$ $(2)$ $\{ m+n\alpha:m\in\mathbb{N},n\in\mathbb{Z}\}.$ $(3)$ $\{ ...
0
votes
1answer
97 views

Calculate power of irrational number modulo of integer

There are very efficient ways to calculate powers of integers modulo an integer, one of them is implemented by the Python pow function. I need to calculate ...
1
vote
1answer
187 views

Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational? [duplicate]

I know the famous proof that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to ...
1
vote
1answer
149 views

Why is $\pi$ irrational?

I've a few questions in mind: Why is $\pi$ irrational? If it is, then how can 2 rational quantities (circumference, diameter) can produce irrational number? How are we able to determine digits of $\...
2
votes
0answers
68 views

Bijection between the set of irrational numbers from 0 to 1 and the set of infinite integers? (Actually, 10-adic integers)

It seems to me we can assign to every irrational from 0 to 1 an infinite whole number in the following way: $$ 0.a_1a_2a_3a_4... \rightarrow ...a_4a_3a_2a_1 $$ For example: $$ 0.14159265... \...
5
votes
1answer
78 views

Does the series $ \sum_k^\infty \frac{k!}{k^k}$ converge to an irrational number and does it have any significance or applications?

I know that the series $$ \sum_k^\infty \frac{k!}{k^k} $$ converges by the ratio test. The sum calculated by wolfram alpha is $~1.87985386217525853348630614507096$ which seems pretty irrational to ...
1
vote
1answer
30 views

a dense set in (0,1)

Define for $\epsilon > 0 $ $$V_\epsilon = \left( \bigcup_{j \in \mathbb{N}} (x_n - \frac{\epsilon}{2^{n+1}} , x_n + \frac{\epsilon}{2^{n+1}}) \right) \ \bigcap \ (0,1)$$ where $x_n$ stems from ...
1
vote
2answers
15 views

Determine the field properties that are satisfied by B, Is B a field?

Let B be the set of all irrational numbers together with the numbers 0, 1, and -1. Let addition and multiplication be defined on B in the same way they are defined for real numbers. Determine the ...
2
votes
2answers
88 views

Prove or give a counterexample If $a \in \Bbb R$\ $\Bbb Q$ exists $n \in \Bbb N$ such that $a^n \in \Bbb Q$

1)If $a \in \Bbb R$\ $\Bbb Q$ exists $n \in \Bbb N$ such that $a^n \in \Bbb Q$ 2)If $a \in \Bbb R$\ $\Bbb Q$ , $_n\sqrt a \in \Bbb R$ \ $\Bbb Q $ $\forall n \in \Bbb N$ For the second one: [by ...
-1
votes
3answers
276 views

Is the sum of two rationals or two irrationals irrational?

1. I know this statement is false (if I am correct) but how to prove it's false? "The sum of two rational numbers is irrational." 2. I know this statement is true (if I am correct) but how to ...
0
votes
1answer
25 views

Sequences of consecutive digits of arbitrary length in irrational ternary numbers (take two).

Suppose we have an irrational number represented in base $3$ such that there can only be a maximum of $n$ consecutive $1$'s or $2$'s in the ternary expansion. Furthermore, suppose the only digit ...
5
votes
5answers
773 views

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

Suppose we have an irrational number represented in base $3$ such that there can only be a maximum of $n$ consecutive $1$'s or $2$'s in the ternary expansion. Does this imply there are arbitrarily ...
5
votes
2answers
45 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. ...
-4
votes
2answers
821 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
100 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
72 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
33 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 ...
3
votes
2answers
167 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. ...
0
votes
1answer
87 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 ...
0
votes
5answers
234 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 \, \sinh\left(\frac{\pi}{2}\right)-\sinh\left(\...
4
votes
1answer
61 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 $$ \lfloor x^k \rfloor \equiv m \pmod{n}? $$
5
votes
2answers
241 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.
6
votes
3answers
277 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
100 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
35 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
3k 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 ...
1
vote
2answers
77 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 ...
1
vote
2answers
81 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
115 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
69 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
81 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 ...