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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2
votes
2answers
118 views

Conjecture: if $a+b$ and $ab$ are rational, $a$ and $b$ are rational

I can't find a rigorous proof but I have a feeling it's true. Informal argument: Suppose $a+b$ and $ab$ are rational, $a$ and $b$ are irrational (since just one can't be irrational). Then $a$ and ...
2
votes
1answer
85 views

Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$

Question : Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$. Hint: My solution doesn't use continued fraction. I am interested in other possible proofs for this question.
9
votes
4answers
1k views

Is the fact that there are more irrational numbers than rational numbers useful?

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of ...
1
vote
2answers
93 views

Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$.

Prove that if $n \geq 2$, then $\sqrt[n]{n}$ is irrational. Hint, show that if $n \geq 2$, then $2^{n} > n$. So, my thought process was that I could show that $2^{n} > n$ using induction, but ...
2
votes
1answer
135 views

Does the limit of a sequence with floor function exist?

Question : Let $a_n=n\alpha-\lfloor n\alpha\rfloor\ (n=1,2,\cdots)$ where $\alpha$ is an irrational number. Then, does the limit $n\to\infty$ of $(a_n)^n$ exist? I know that ...
1
vote
1answer
110 views

How to prove that $\cos(n)$ is irrational?

We know that $\cos(1)$ is real and transcendental (1). Then by using the fact that for every $n \in \mathbb{N}$ there exists a polynomial $P_n$ of degree $n$ with integer coefficients such that ...
3
votes
1answer
89 views

Irrationality measure.

I would like someone to give me a definition of what irrationality measure is, I have stumbled over several definitions which may be equivalent but as I lack understanding I cant see this correlation. ...
15
votes
1answer
322 views

Can we prove that the solutions of $\int_0^y \sin(\sin(x)) dx =1$ are irrational?

Can we prove that the solutions of $$\int_0^y \sin(\sin(x)) dx =1$$ are irrational? Wolfram Alpha gives two approximate sets of solutions as $\{4.58+2\pi k|k\in\mathbb{Z}\}$ and $\{1.69+2\pi ...
4
votes
3answers
120 views

Real numbers that are not the roots of any polynomial equation with algebraic coefficients

An algebraic number is a number which is a root of some non-zero polynomial equation with rational coefficients. A transcendental number is a number which is not a root of any non-zero polynomial ...
1
vote
3answers
58 views

negative powers $(x^{-2} = 1/x^2)$

I need clarification for negative power of a number. I understand $x$ to the power of $2$ is equal to $x\cdot x$ But how $x$ to the power of $-2$ is equal to $\dfrac{1}{x^2}$ ?
0
votes
0answers
32 views

Integer algorithm

I have this equation: $$\ln_p y = x+\ln_k z$$ for $p, y, x, k, z \in \mathbb{N}$ Now consider that I have the values for $y$ and I can generate in anyway possible, the value for $x$. How would i ...
1
vote
0answers
51 views

“Building blocks” for computable functions

In an (otherwise very enlightening) answer to another question of mine the question came up What functions are allowed as building blocks for computable functions? I was astonished that there ...
4
votes
4answers
486 views

Why there are irrational numbers?

I do not quite get it. Why can't we represent all real numbers as a sum of rational numbers? Why do we need irrational numbers? For example, ...
2
votes
1answer
80 views

is $(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}))\cup((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q})$ connected? path connected?

let $$X=(\mathbb{Q} \times (\mathbb{R}\setminus\mathbb{Q}) ) \cup ((\mathbb{R}\setminus\mathbb{Q})\times\mathbb{Q}) $$ and let $$\tau=\tau (\text{euclid})$$ what are the connected components of ...
2
votes
1answer
41 views

Solving surds without compairing

Question: Let $a + \sqrt{2b} = 3 - 2\sqrt{2}$ .Find the value of $a - \sqrt{2b}$ What I did: I compared the whole numbers and the irrational numbers in both sides and calculated the answer $3 + ...
2
votes
2answers
178 views

A dense set on $[0,1)$

Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. Can anyone give some hint to solve this problem? I tried ...
2
votes
1answer
37 views

Need help to simplify irrational equation

I have faced a problem simplifying this equation. . I tried to solve it this way: , but I just can't get the correct answer. This equation is from high school course, so it must have quite a simple ...
0
votes
3answers
49 views

A pretty much simple number theory problem

Let $x$ be an irrational number, and $n$ be a positive integer. Will there ever be a set of $(n,x)$ which satisfies $x(n-x) \in \mathbb{Z}$ ? If so, could you suggest those numbers? And, if not, ...
0
votes
1answer
48 views

Square root of an odd composite being irrational

Is there an odd composite number $n$ such that $\sqrt{n}$ is irrational?
4
votes
6answers
428 views

Is $i$ irrational?

On the one hand, $i(=\sqrt{-1})$ cannot be expressed as a ratio of integers, so, by definition, $i$ is not rational $\iff i$ is irrational. However, the set of irrational numbers, ...
5
votes
2answers
185 views

A question about decimal representation of irrational numbers.

Is this true that any finite word of the alphabet $\mathcal{A_9}=\{0,1,2, \ldots,8,9\}$ appears somewhere in the decimal representation of $\sqrt{2}$ ? Thanks !
8
votes
0answers
129 views

The “trick” functions in the “$\pi$ is transcendental” proofs

I was reading this paper and I wondered how did Hermite decide to define a function $$f(x)=\frac{x^{p-1}(x-1)^p\cdots (x-m)^p}{(p-1)!}$$ Are these functions only tricks or there is a deeper meaning?
16
votes
10answers
3k views

Visualizing the square root of 2

A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here. We were talking about how the square root of 2 is an ...
3
votes
3answers
292 views

Prove $\sqrt6$ is irrational

Suppose $\sqrt6 = \frac pq$ where $p$ and $q$ have no common factors. $$6 = \frac {p^2}{q^2}$$ $$6p^2 = q^2$$ So $q^2$ and therefore $q$ is divisible by $6$. $$p^2 = \frac {q^2}{6}$$ So $p^2$ ...
7
votes
1answer
244 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
4
votes
4answers
194 views

Prove that $(4/5)^{\frac{4}{5}}$ is irrational

Prove that $(4/5)^{\frac{4}{5}}$ is irrational. My proof so far: Suppose for contradiction that $(4/5)^{\frac{4}{5}}$ is rational. Then $(4/5)^{\frac{4}{5}}$=$\dfrac{p}{q}$, where $p$,$q$ are ...
3
votes
1answer
64 views

Euclidean geometry and irrational numbers.

I was wondering, given a square that is $1 \times 1$, how can we know that the diagonal is an irrational length geometrically??? We could use the Pythagorean Theorem to see that the diagonal of a ...
0
votes
1answer
44 views

Looking for irrational Numbers Proof [duplicate]

$a,b,c,~$and $d$ are rational numbers. $b>0$ and $d>0$ the $\sqrt{b}$ and the $\sqrt{d}$ are both irrational. if $a+\sqrt{b}=c+\sqrt{d}$ show that $a = c$ and $b = d$. I know that a=c and ...
3
votes
3answers
186 views

Is it possible that $\pi$ is finite in other numerical bases?

In base $\pi$, the number $\pi$ is $1\cdot \pi^1 + 0\cdot \pi ^ 0 $, which is equal to $10$. So, is $\pi$ an irrational number in all bases or not?
0
votes
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
45 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
234 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
113 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
237 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
142 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
563 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
125 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
37 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
206 views

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

How would you prove that $\sqrt[n]{2}$ is irrational?, where $n \in \{2, 3, 4, \ldots\}$.
0
votes
6answers
160 views

Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational.

The sum of a rational and irrational number is always irrational, that much I know - thus, if n is a perfect square, we are finished. However, is it not possible that the sum of two irrational numbers ...
3
votes
2answers
126 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
41 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
143 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
268 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 ...