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

4
votes
2answers
85 views

To prove that element $\frac{3}{n}+i\frac{4}{5}$ has an infinite order in $\mathbb{C}$ for any $n\in\mathbb{Z}\backslash \{0\}$

The problem is to prove that element $z=\frac{3}{n}+i\frac{4}{5}$ has an infinite order in the group $(\mathbb{C},\, \cdot\, )$ for any non-zero integer $n$. Let's consider the case $|n|\neq 5$. The ...
0
votes
1answer
68 views

What is the closest fraction (that isn't something like 31415…/1000…) that gets you pretty close to pi?

I'm just wondering what is the closest fraction (question for math nerds and geniuses) that isn't like pi/length-of-pi that gets you relatively close (like accurate to the 20th place) to pi? For ...
1
vote
3answers
78 views

Prove $\sqrt{2}+\sqrt[3]{5}$ is irrational [duplicate]

Prove: $\sqrt{2}+\sqrt[3]{5}$ is irrational I have tried to look at $1+\sqrt{2}+\sqrt[3]{5}$ but I can see how to continue
2
votes
3answers
198 views

Prove that $\sqrt{45}$ is irrational (using Euclid’s Lemma)

Prove that $\sqrt{45}$ is irrational (using Euclid’s Lemma) Assume $\sqrt{45}$ is rational. By definition of rational : $\sqrt{p}=\frac{a}{b}$= $\sqrt{45}$=$\frac{a}{b}$ for some $a,b$ integers and $...
-1
votes
2answers
162 views

Proving $\sqrt[4]{4}$ is irrational

Prove: $\sqrt[4]{4}$ is irrational I know that $\sqrt{p}$ is irrational where $p$ is a prime number. So $\sqrt[4]{4}=\sqrt[4]{2*2}=16*\sqrt[4]{2}=16*2^{\frac{1}{2}^\frac{1}{2}}$ What can I say ...
0
votes
1answer
66 views

determine the frontier

determine the frontier of the set R\Q (where R is the real numbers and Q is the rational numbers). I figured R\Q is the same as saying the real line minus all the rational numbers which would just ...
0
votes
0answers
20 views

Check the validity of a statement concerning functional continuity and limits at infinity, perhaps also involving a bit of real-number analysis [duplicate]

I've been stuck on this problem: If $f\in C(0,+\infty)$, and $f(na)\to 0$ as $n\to +\infty$ for every $a>0$, then is $$\lim_{x\to+\infty}f(x)=0$$ also true? If we replace continuity by ...
1
vote
4answers
105 views

Proof that this is an irrational number

Prove that $2\sqrt2 + \sqrt7 $ is an irrational number. I am trying to use contradiction to show that this is irrational. Also I am using the fact that $ 2\sqrt2 + \sqrt7 = \frac{1}{2\sqrt2 - \sqrt7}$....
3
votes
2answers
63 views

Prove that $\sqrt{5}+1$ is irrational

I'm guessing this is pretty basic, but I've been struggling with this for a while now. Any help would be appreciated. Thanks.
-1
votes
2answers
89 views

Why a segment of length $\sqrt{2}$ can be drawn but a segment of length $\pi$ cannot?

We know that both $\pi$ and $\sqrt{2}$ are irrational. Also, it has been proved that a segment of length $\pi$ can not be drawn whereas a segment of length $\sqrt{2}$ can be drawn. Why is it so, ...
1
vote
1answer
18 views

A question of rationality of integral powers

Given $a,b\in\mathbb{Z}$, is $x$ from $a^x=b$ ever rational? More specifically, is $x$ in $2^x=3$ irrational?
0
votes
0answers
26 views

Finding a 10-digit pattern in the decimal expansion of a normal number?

Is there any way to estimate how "deep" we'd need to go into the decimal expansion of a normal number to find a specific string of digits, say a 10-digit pattern? What about a 100-digit pattern?
3
votes
2answers
97 views

Does $\sin n$ have a maximum value for natural number $n$?

In formal, does there exist $k\in\mathbb{N}$ such that $\sin n\leq\sin k$ for all $n\in\mathbb{N}$?
5
votes
2answers
156 views

Is $\sqrt[2]{(2/7)}$ irrational?

I have to show that the $\sqrt(2/7)$ is irrational. Here is my work.
1
vote
1answer
105 views

Neighbors of Irrational Numbers on Real Number Line

I was looking at a post on MathOverflow about "What is your favorite 'strange' function?" One of the answers mentioned Thomae's function claiming that the function was "continuous at all irrationals ...
1
vote
1answer
35 views

Characterizing the roots of rational numbers

I am trying to prove the statement: if $n \in \mathbb{Q}$ and $\sqrt[m]{n} \in \mathbb{Q}$ for all positive integers, then $n = 1$. In my work, I have done all the work given by the top answer to ...
1
vote
1answer
105 views

Are irrational numbers irrational by nature? [duplicate]

I remember hearing an interesting theory once, I don't know the source. Since there are some numbers that are precisely expressible in decimal notation that repeat in a binary base, and vice versa, ...
2
votes
1answer
42 views

How to obtain the negative rational numbers

Is there any formula for generating negative rational numbers? Can Calkin–Wilf tree be used for negative numbers?
1
vote
3answers
101 views

why is the value of $(-1)^\frac{2}{3}$ not 1?

$(-1)^2$ equals 1 also $(-1)^\frac{1}{3}$ equals -1 any way you arrange it, it should be one. how ever, typing this into Wolfram or google,gives me an irrational complex number. I would love some ...
28
votes
10answers
4k views

Is it possible to represent every irrational number as a (limit of) an infinite sum of rational numbers? [duplicate]

For instance, we can certainly represent π in this fashion. $$ \frac{\pi}{4} \;=\; \sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} .\! $$ $\ln(2)$ is also irrational. And even that can be represented as an ...
0
votes
5answers
96 views

Prove rational sum and product of two irrational numbers

I need to prove that $$\exists a,b \in \mathbb{R} \setminus \mathbb{Q} : a + b, ab \in \mathbb{Q}$$ Any ideas? I, unfortunately, don't have one yet. The most obvious way with equations in integers (...
-1
votes
1answer
134 views

If $x$ is an irrational number, then $3x^2 + 2$ is an irrational number? [closed]

Prove or disprove using direct or indirect proof that if $x$ is irrational, then $3x^2 + 2$ is irrational? Thank you in advance.
-3
votes
4answers
270 views

Is $\pi^0$ actually rational? How about $\pi^i$? [duplicate]

Is there a rational argument that a transcendental or irrational number raised to zero should magically turn it into an integer, beyond obtuse convention? How about $\pi^i$? Is there a reasonable ...
2
votes
0answers
59 views

Alternative prrof of $\sqrt 2 $ (and $\sqrt {\text {of any non square integer}}$ ) not being rational

Long time ago for an assignment, I submitted an alternative proof of $\sqrt 2$ not being rational along the following lines: suppose $\frac{a^2}{b^2}=2$ then we must also have $a=b+n$ for $a,b,n$ ...
-1
votes
2answers
122 views

Prove that $\sqrt{2+\sqrt 3}$ is irrational [closed]

Prove that $\sqrt{2+\sqrt3}$ is irrational. I can't seem to figure this one out.
0
votes
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
62 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
69 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
227 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
130 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
64 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
763 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
97 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
289 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
582 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
152 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
380 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
148 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
45 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
61 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
94 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
177 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
147 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 ...