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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1answer
37 views

Is there any combination of numbers which upon division gives the exact number of P?

In other words, there are (probably) infinite combination of numbers/operations which leads to irrational numbers. So I wonder, if there is one which gives exact number representation of P(π)? Do we ...
5
votes
1answer
60 views

$\lim \{r^n\}$ exists, Is $r$ an integer?

$r\in\Bbb R$, $|r|\gt1$ and $\lim\limits_{n\to\infty}\{r^n\}$ exists. Can one conclude that $r$ is an integer? Here, $\{x\}=x-[x] $ is the fractional part of $x\in\Bbb R$ If $r\in\Bbb Q$, the ...
0
votes
1answer
47 views

Is Dirichlet's function enough to prove constants like $\gamma$ irrational?

This function appears without any reference in the book The Irrationals : $$\lim_{m\to\infty}\lim_{n\to\infty} \cos^{2n}(m!\pi x)=\left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q}\\ ...
8
votes
3answers
245 views

If $\sum\frac1{a_n}$ is convergent, then irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=1$$ If $\sum\limits_{n=1}^\infty\frac1{a_n}$ is convergent, can one conclude ...
2
votes
1answer
114 views

Do circles exist

So I was wondering about circles today and if they really do exsist. If you graph a circle in function mode, your equation looks like$$y=\sqrt{1-x^2}$$ Now for simple purposes lets take a portion of ...
17
votes
3answers
241 views

Is $\sum\limits_{n=1}^\infty\frac1{a_n}$ irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=+\infty$$ Can one conclude that $\sum\limits_{n=1}^\infty\frac1{a_n}$ is an ...
9
votes
1answer
144 views

If $(a_n)$ is increasing and $\lim_{n\to\infty}\frac{a_{n+1}}{a_1\dotsb a_n}=+\infty$ then $\sum\limits_{n=1}^\infty\frac1{a_n}$ is irrational

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{a_1a_2\dotsb a_n}=+\infty$$ then $\sum\limits_{n=1}^\infty\frac1{a_n}$ is an irrational ...
0
votes
1answer
24 views

Help proving a theorem in my textbook

If $r \in \mathbb{N}$ is not a perfect square, then $\sqrt{r}$ is irrational. For reference, an integer $n$ is a perfect square if $n=m^2$ for some $m \in \mathbb{Z}$. Any help proving this ...
2
votes
2answers
48 views

Help with a proof my professor gave my class

Let $x,y \in \mathbb{R}$ with $x<y$. There exists an irrational number $z$ such that $x<z<y$. My proof so far: Let $x,y \in \mathbb{R}$ and assume $x<y$. Then, by Theorem 11.8 (in ...
14
votes
3answers
568 views

Irrationality of $\pi$ another proof

Proposition. Let $\alpha\in\mathbb{R}$. If there is a sequence of integers $a_n,b_n$ such that $0<|b_n\alpha-a_n|\longrightarrow 0^+$ as $n\longrightarrow \infty$, then $\alpha$ is irrational. ...
1
vote
1answer
57 views

Irrationality Measure $x\in \mathbb{Q} \Longleftrightarrow \mu(x)=1$

Let $x$ be a real number, and let $R$ be the set of positive real numbers $\mu$ for which $$0<|x-\frac{p}{q}|<\frac{1}{q^{\mu}}$$ has (at most) finitely many solutions $p/q$ for $p$ and $q$ ...
2
votes
3answers
130 views

Help with a proof that sequence of rational numbers $ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$ converges to an irrational, $\sqrt2$

I know that there are sequences of rational numbers with irrational limits. One in particular I've seen is $$ a_n = \frac {a_{n-1} + \frac {2}{a_{n-1}}}{2}$$ with $a_0 =1$, This is clearly rational ...
0
votes
1answer
39 views

Irrationality of e and farey fractions

How do we go about proving that $$[k! e] = k! \sum_{j=0 -> k} \frac{1}{j!}$$ I know that we could write $$e = \sum_{j=0 -> \infty} \frac{1}{j!}$$ But I don't see how that's going to help in ...
2
votes
5answers
208 views

How would you prove that $\sqrt[n]{2}$ is irrational?

How would you prove that $\sqrt[n]{2}$ is irrational?, where $n \in \{2, 3, 4, \ldots\}$.
3
votes
2answers
130 views

Understanding proof that $\pi$ is irrational

Reading this: Simple proof that $\pi$ is irrational, I fail to understand the following part: Since $n!f(x)$ has integral coefficients and terms in $x$ of degree not less than $n$, $f(x)$ and ...
0
votes
1answer
42 views

Farey sequence problem with irrational numbers

If an irrational number $\theta$ lies between two consecutive terms $a/b$ and $c/d$ of the Farey sequence of order n, prove that at least one of the following holds: $|\theta- a/b| < 1/2b^2$ or ...
3
votes
3answers
145 views

Check that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational

How to prove that $\sqrt{7}(\sqrt[3]{5} - \sqrt[5]{3})$ is not rational. I will appreciate any proof, but I had such exercise during lecture in field theory. Thanks.
3
votes
2answers
83 views

How to prove that: $\sqrt{25!+3} \in \mathbb{R}\setminus\mathbb{Q}$

How can I prove that: $$\sqrt{25!+3} \in \mathbb{R}\setminus\mathbb{Q}?$$ Thanks!
4
votes
5answers
273 views

Prove the irrationality of $0.235711131719…$

How can I prove that the number formed by concatenating the primes in order i.e. $0.235711131719...$ is irrational. I know I have to demonstrate that it has no period, but I'll be so thankful if ...
3
votes
1answer
67 views

Cantor's proof on uncountability of irrationals

I have a question regarding Cantor's proof that the set of irrational numbers is uncountable. As far as I know, to prove this, Cantor proves that there exists a mapping between irrationals and ...
5
votes
2answers
504 views

Is the Nested Radical Constant rational or irrational?

Given the sequence $A_n=\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{\dots+\sqrt{n}}}}}$: Are there any known rational elements in $A_n$, or has it been proved that all are irrational? Is there any proof for ...
2
votes
1answer
114 views

Irrational fractional parts - patterns

Some fractional-part list plots are: $\text{listplot of }|[\pi x]-\pi x|\text{, for }x \in \mathbb{Z} \text{ and } \text{listplot of }|[ex]-ex|\text{, for }x \in \mathbb{Z}$ $\text{listplot of ...
2
votes
0answers
83 views

Question about $e^{e^{e^e}}$

Is there a proof that the power tower of length $4$ of $e$ is irrational? Is it known whether or not $$e^{e^{e^e}}$$ is transcendental?
8
votes
2answers
249 views

Is $\ \Large\pi^e$ rational?

Is the number $\ \Large\pi^e$ rational?
6
votes
3answers
201 views

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$.

Question : Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions ...
2
votes
2answers
87 views

Prove that if $y,z \in\Bbb Q$ then $y^z \in\Bbb A$

Question : Prove that if $y,z \in\Bbb Q$ then $y^z \in\Bbb A$. My attempt: Definition 2.7.8 states that a number $s$ is an algebraic number when there exists some $p \in\Bbb Z[x]$ such that ...
3
votes
3answers
166 views

How can you proof that the sum of three roots is irrational?

I would like to know how to prove that $\sqrt{2} + \sqrt{5} + \sqrt{7}$ is an irrational number. I know how to do the proof for a sum of two roots. Can I just define $\sqrt{2} + \sqrt{5} :=c$ and then ...
3
votes
5answers
125 views

$\sqrt[3]{2}$ is not the root of a quadratic with rational coefficients?

How can one show that $\sqrt[3]{2}$ is not a root of a quadratic with rational coefficients? It is clear that if $\sqrt[3]{2}$ is the root of such a quadratic, then it is also the root of a quadratic ...
1
vote
1answer
126 views

2.71828. And then another 1828.

This may qualify as the silliest math.SE question ever, but am I really the first person ever to worry about this? The decimal expansion of $e$ has a 2. And then a 7. And then a 1828. And...well, then ...
3
votes
2answers
98 views

Values $nx - [nx]$ are distinct for an irrational number

When reading the proof for Dirichlet's Approximation Theorem, I came across the following statement: If $x$ is irrational, then $nx - [nx]$ are distinct for all $n \in \mathbb{Z}$. I don't ...
0
votes
3answers
248 views

Is there any geometry where ratio of circle's circumference to its diameter is rational?

In Euclidean geometry, the ratio of the circumference of a circle to its diameter is an irrational number, 3.14159 and so on. But if you change to non-Euclidean geometries, you get other values for ...
0
votes
1answer
38 views

If $\frac1\alpha+\frac1\beta=1$, irrational, then $\{\lfloor n\alpha\rfloor:n\in\Bbb N\}\uplus\{\lfloor n\beta\rfloor:n\in\Bbb N\}=\Bbb N$

Let $\alpha,\beta\in\Bbb R\setminus\Bbb Q$ such that $\frac1\alpha+\frac1\beta=1$, and define $S(x)=\{\lfloor nx\rfloor:n\in\Bbb N\}$. (Note that my convention takes $0\notin\Bbb N$.) The claim is ...
8
votes
3answers
133 views

How to understand “Union of balls centered at rational numbers is way less than $\mathbb{R}$

A few month ago I had to prove $\lambda(\mathbb{Q}) = 0$ (where $\lambda$ is the one-dimensional Lebesgue measure). The idea: Let $\varepsilon \gt 0, r_n := \frac{\varepsilon}{2^n}$ and $\mathbb{Q} = ...
1
vote
1answer
55 views

Rational vs irrational

If two points on a number line is shown, are rational numbers between the two points is more or irrational number is more ? I have tried using probability , my collegue who was like my teacher also ...
1
vote
1answer
192 views

Dense sequence in $[0,1]$

There is the theorem proved by Liouville which states that if $x$ is irrational then there are infinitely many fractions $\frac{p}{q}$ such that $|x-\frac{p}{q}|<\frac{1}{q^2}$, i.e. ...
4
votes
1answer
144 views

Do rational and irrational numbers flip-flop?

I have found out that between every 2 rational numbers there is an irrational number, and between every 2 irrational numbers, there is a rational number. Does this mean that the rational and ...
2
votes
1answer
431 views

Irrationality of $\pi$ and circumference to diameter ratio.

How is $\pi$ actually defined? If it is defined as the ratio of the circumference of a circle to its diameter then from this definition itself either of the circumference and diameter has to be ...
0
votes
1answer
61 views

rational numbers

It is by the rule that a rational number can be expressed in P/Q form and an irrational number cannot be expressed in such form. It is even said that (in order to promote the P/Q form for expressing ...
5
votes
4answers
429 views

William Lowell Putnam Integral Problem [closed]

Prove That $$ \frac{22}{7}-\pi = \int_{0}^{1}\frac{x^{4}\left(1 - x\right)^{4}}{1 + x^{2}}\,{\rm d}x $$
2
votes
1answer
104 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 ...
4
votes
2answers
401 views

Generalizations of the golden and silver ratios, and their significance

$\Phi$, or the golden ratio, is basically $\frac{a+b}{a}=\frac{a}{b}$. The silver ratio corresponds to a similar idea of: $\frac{2a+b}{a}=\frac{a}{b}$. I've read on Wikipedia that both of these ratios ...
0
votes
0answers
82 views

irrational, proving or finding counterexapmle [duplicate]

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational. Can anyone give me a hint just about how to start cause i ...
0
votes
5answers
553 views

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational.

The following is a Homework Question that I've been working on and I would like some feedback on my answer: Prove or find a counterexample: For all real numbers $x$ and $y$ it holds that $x + y$ is ...
21
votes
4answers
2k views

Can every irrational number be written in terms of finitely many rational numbers?

Consider the irrational number $\sqrt{2}$. It can be written in terms, i.e., in a closed form expression, of two rational numbers as $2^{\frac{1}{2}}$. Does it hold in general that every irrational ...
4
votes
1answer
74 views

Are there numbers that if proven rational (or irrational) will have important consequences to mathematics?

We see all the time conjectures and proofs that specific (real) numbers are (more often than not) irrational. I'm wondering that apart from the mathematical curiosity motivating such proof attempts, ...
2
votes
2answers
213 views

Let a, b, c, d be rational numbers… [closed]

Let $a, b, c, d$ be rational numbers, where $\sqrt{b}$ and $\sqrt{d}$ exist and are irrational. If $a + \sqrt{b} = c + \sqrt{d}$, prove that $a=c$ and $b=d$.
6
votes
1answer
207 views

Is $\log 2\pi$ rational?

Is it known whether $\log 2\pi$ is rational (where the base of the logarithm is $e$)? Or algebraic?
1
vote
1answer
80 views

Confusing rational numbers

Question: If $$x = \frac{4\sqrt{2}}{\sqrt{2}+1}$$ Then find value of, $$\frac{1}{\sqrt{2}}*(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x - 2\sqrt{2}})$$ My approach: I rationalized the value of $x$ to ...
2
votes
1answer
53 views

Logic verification: $x^3$ is irrational, then $x$ is also irrational

Prove, by contraposition, if $x^3$ is irrational, then $x$ is also irrational. Just a verification do I need to show that given $x$ is rational $x^3$ is also rational? Suppose $x \in \mathbb{Q}$ ...
5
votes
0answers
152 views

Are there integers $a, b$ s.th. $\pi^a = e^b$?

Is $\log \pi $ a rational number? That is, are there non-zero integers $a, b$ s.th. $\pi^a = e^b$ ?