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.

learn more… | top users | synonyms

0
votes
2answers
91 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
266 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 ...
25
votes
2answers
493 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: $$ ...
2
votes
1answer
46 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
138 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
321 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
132 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 ...
0
votes
1answer
33 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
186 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
56 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
70 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
163 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
142 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
63 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
73 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
28 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
84 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
236 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
24 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
766 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
44 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
692 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
94 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
29 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
156 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
82 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
226 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
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
207 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
271 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
88 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
34 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
2k 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
105 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
61 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
80 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
100 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 ...
-6
votes
2answers
79 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.
19
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
82 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
88 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
35 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 ...
14
votes
1answer
154 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
27 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
48 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 ...