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

1
vote
1answer
35 views

Continous function from $ \Bbb Q \rightarrow \Bbb R $, $ f = 1 $ for $x > \sqrt2$ and $ f = 0$ for $x < \sqrt2$

I'm not really sure how to go about this problem. Show that $h : \Bbb Q \rightarrow \Bbb R $, with $$ h(x)=\begin{cases} 0 &\text{for $|x|< \sqrt{2}$} \\ 1 &\text{for $|x|>\sqrt{2}$} ...
1
vote
1answer
81 views

Which set is more dense: set of irrational numbers or set of rational numbers? [duplicate]

Is the infinity of irrational numbers equal to the infinity of rational numbers? Or is one is greater than other? And what is the proof? I could not find out a rigorous proof about this. P.S. I am ...
1
vote
1answer
59 views

Set of Rational numbers a countable set?

How can we say that rational numbers is a countable set? I can divide a rational number by infinite different number of natural numbers so shouldn't there be infinite rational numbers. ...
0
votes
0answers
37 views

Arithmetic geometric mean - irrational, algebraic, trancendental?

Are there some general theorems about rationality/irrationality and abgebraicity/transcedentality of arithmetic-geometric mean? At least for some group of numbers (like natural numbers)? Or even for ...
2
votes
2answers
88 views

The irrationality of $\sqrt[n]{2}$ from the FLT.

It's common to see the Fermat Last Theorem being used to prove the irrationality of $\sqrt[n]{2}$. In fact, according this post, the said proof appeared in American Mathematical Monthly. On the other ...
0
votes
2answers
46 views

Given a rational number and an irrational number, both greater than 0, prove that the product between them is irrational.

Does this proof I made make sense? Proof// $\mathbf a$ is the rational number, $\mathbf b$ is the irrational number. Assume that $\mathbf {a * b}$ is rational due to proof by contradiction. ...
1
vote
1answer
66 views

Geometric proof for irrationality of $\pi$

Is there a geometric proof for irrationality of $\pi$? That would be neat.
0
votes
0answers
36 views

How to prove $a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+…+a_{n}\sqrt[b_{n}]{c_{n}}$ is irrational?

Let's define the number $$A=a_{1}\sqrt[b_{1}]{c_{1}}+a_{2}\sqrt[b_{2}]{c_{2}}+.....+a_{n}\sqrt[b_{n}]{c_{n}}$$ where $a_{1}, a_{2}, ..., a_{n}$ are positive integers and $b_{1}, b_{2}, ..., b_{n}, ...
3
votes
2answers
108 views

Elementary proof that $\pi$ is irrational

I'm trying to understand the first proof in this page. So we have $$S=\frac{\pi }{4}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}=S_{n}+R_{n}$$ where $S_{n}=\sum_{k=1}^{n}\frac{(-1)^{k-1}}{2k-1}$ and ...
8
votes
0answers
124 views

Swapping the digits of an algebraic number (e.g. $\sqrt 2$)

Let an algebraic number, say $ a=\sqrt 2 = 1.41421356237309504880...$, and define $$b=f(a)=1.14243165323790058408...$$ by swapping the digits $a_{2i+1}$ and $a_{2i+2}$ for $i≥0$, corresponding to ...
0
votes
1answer
24 views

Shift of finitely many reals into irrationals

This question came in one of class-room discussion. Given finitely many distinct real numbers $x_1,x_2,\cdots, x_n$, does there exists a real number $y$ such that $y+x_1, y+x_2, \cdots, y+x_n$ are ...
5
votes
5answers
131 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
0
votes
0answers
60 views

Why are positive rational numbers countable but real numbers are not? [duplicate]

If we can say that any positive rational number is countable or listable by showing that every positive rational number is the quotient of p/q of two positive ...
0
votes
1answer
14 views

Additivity Implies Homogeneity of Rational Scalars

I did my best to search for this question on the site- but I did not find it. Here it is: If a function $f:\mathbb{R}^2\to\mathbb{R}$ satisfies $f(u+v)=f(u)+f(v)$ for all $u,v\in\mathbb{R}^2$, ...
4
votes
1answer
100 views

Is $\arctan2$ irrational? [duplicate]

Is $\tan^{-1}2$ an irrational number or a rational number? How to show that? Or generally how to show $\tan^{-1}3, \tan^{-1}4, \tan^{-1}5...$ is irrational or rational?
2
votes
2answers
54 views

Square root of the product of consecutive natural numbers is irrational

Prove that for all $n\in\mathbb{N}$ the number $\sqrt{n(n+1)}$ is irrational. My first move would be: Let's assume that it's not, that it $\sqrt{n(n+1)} = \frac{a}{b}$, where $a,b\in\mathbb{N}$ and ...
3
votes
3answers
80 views

For what numbers $n$ is $\sqrt{n}$ irrational?

I would say it has something to do with the numbers that can be expressed as a factor of different prime numbers, but when I get to $8$, that can be changed to $2^3$, which goes against this. Is there ...
5
votes
2answers
88 views

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$? I know that \begin{align*} \mathbb{Q}(\sqrt{2}) &= \{a+b\sqrt{2} \mid a,b \in \mathbb{Q}\}, \\ \mathbb{Q}(i) &= ...
0
votes
1answer
35 views

$x^J = y$, $J = 2.455\ldots$ What's the rest of $J$?

I have a problem where I need to know what J is. I do x^J and get y. For example, if I do 5^J, I would want to get 55 as y. Same with 4^J = 30. When J is 2.455, it works up to 4 only! I need for ...
1
vote
1answer
34 views

Why are surds put on the numerator in the final answer when it is a fraction. [duplicate]

I have learnt that a fraction with a surd in its most simplest form should have the surd in the numerator and not the denominator? Why is it convention not to leave the surd on the denominator? Is it ...
2
votes
1answer
71 views

Proof of irrationality of $\zeta(2)$ without explicitly calculating it

Question is pretty much the title. It is pretty easy to show that $\zeta(2n)$ is irrational for all $n$ once you know that $\zeta(2n)$ is a rational multiple of $\pi^{2n}$ (and then also use the fact ...
7
votes
3answers
96 views

Is there an irrational number $a$ such that $a^a$ is rational?

It can be proved that there are two irrational numbers $a$ and $b$ such that $a^b$ is rational (see Can an irrational number raised to an irrational power be rational?) and that for each irrational ...
2
votes
1answer
45 views

$\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$

I'd appreciate if you could please check to see if my proof is valid. Find $\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$. Solution: $q^2 > p^2/5 \iff q > ...
11
votes
2answers
198 views

Is $x$ irrational when $2^{x}+3^{x}=6$?

Is $x$ rational or irrational when $2^{x}+3^{x}=6$. How to show that?
0
votes
2answers
36 views

How do you find the intersection of the rational numbers, and in interval of irrational numbers?

Take for example $Q ∩ [ - \sqrt(2) , \sqrt (2)]$? Would it be $[ - \sqrt(2) , \sqrt (2)]$ or is this untrue since they are not in $Q$?
2
votes
1answer
74 views

Is $0.\overline{0}1$ a valid repeating decimal? [duplicate]

Surprisingly, I never came across a repeating decimal, which did not include the last digit, so I'm wondering if this would even be a valid notation? $$0.\overline{0}1$$ So the following statement ...
2
votes
1answer
49 views

How can all of them be irrational ??

Assume that $\{x,y,x^2,y^2,xy\}$ are all irrational. Can it be true that all of $\{x-y,x+y,x^2-y^2,x^2+y^2\}$ are irrational? Details: $|x|\ne|y|$ and $x,y\in\mathbb R$. In the ...
0
votes
2answers
46 views

How can we show the inequality?

Let $h_1,h_2:[0,1]\rightarrow \mathbb{R}$ be continuous functions. How can we show that $h_1(x)\leq h_2(x)$ for each $x\in [0,1]$, given that $h_1(x) \leq h_2(x)$ for each $x\in \mathbb{Q}\cap ...
0
votes
2answers
42 views

Rational mean of irrational numbers?

My teacher tells me that in the vicinity of any rational number, an irrational exists. To elucidate, I presume, he further went on to say, if a function, if defined to give 1 for every rational number ...
1
vote
3answers
67 views

Proving that there is no continuous function $f:\Bbb R\to\Bbb R$ satisfying $f(\Bbb Q)\subset\Bbb R-\Bbb Q$ and $f(\Bbb R-\Bbb Q) \subset\Bbb Q$. [duplicate]

How can I prove that there is no continuous function $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(\mathbb{Q}) \subset \mathbb{R}\backslash \mathbb{Q}$ and $f(\mathbb{R}\backslash \mathbb{Q} ) \subset ...
5
votes
3answers
110 views

Find the third rational point on the curve: $y^2 = x^3 + 8$

I am trying to find a third rational point on the curve $y^2 = x^3 + 8$ According to the my professor's solution, the idea is to find two rational points then solve for the third point. These are ...
3
votes
1answer
91 views

Is $\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+…$ irrational?

Is there known way to determine whether the infinite sum below is rational or not? $$\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+...$$
0
votes
3answers
68 views

Find all rational points where $x^2 - y^2 = 1$ (need help simplifying quadratic formula) [duplicate]

The original problem is to find all rational points where $x^2 - y^2 = 1$ I know how to go about the problem, but whenever I get to the point of simplifying my equation, I keep having problems. This ...
2
votes
3answers
131 views

Find a formula for all the points on the hyperbola $x^2 - y^2 = 1$? whose coordinates are rational numbers.

So, I know that we first need to have an initial point. The answers I have say it's $(-1, 0)$ which makes sense because it satisfies the equation. But for example $(1, 0)$ satisfies it too. Why did we ...
20
votes
9answers
2k views

How to find irrational numbers between rationals. (And is my method correct?)

I have a question from an A-level revision book: Find an irrational number which lies between $\frac34$ and $\frac78$. What is the correct method for doing this? Here is my method: Square ...
2
votes
0answers
23 views

Is $\sum \frac{n+1}{b_{n+1}}$ irrational, when $b_1=2$ and $b_{k+1}=2^kb_k(b_k-1)+1$, $k\geq 1$?

Let the sequences of positive integers $$a_n=n$$ when $n\geq 1$, and $$b_{n+1}=2^nb_n(b_n-1)+1$$ for $n\geq 1$ taking $b_1=2$. I've computed with previous sequences to assert that satisfy the ...
1
vote
2answers
45 views

Density of positive multiples of an irrational number

Let $x$ be irrational. Use $\{r\}$ to denote the fractional part of $r$: $\{r\} = r - \lfloor r \rfloor$. I know how to prove that the following set is dense in $[0,1]$: $$\{\{nx\} : n \in ...
2
votes
2answers
61 views

Is the ratio of two natural logarithms irrational or rational?

Is there any way to prove that the ratio of two natural logarithms is rational or irrational? Take the natural logarithms of $a = 25$ and $b = 6$, for example. Can you prove $\ln(a)/\ln(b)$ rational ...
7
votes
1answer
176 views

Are $\frac{\pi}{e}$ or $\frac{e}{\pi}$ irrational?

Is it clear whether $\displaystyle \frac{\pi}{e}$ or $\displaystyle \frac{e}{\pi}$ are irrational or not? If not, then there would exist $q,p\in \mathbb{Z}$ such that $$p\cdot \pi = q\cdot e$$
1
vote
1answer
49 views

Is there an irrational number arbitrarily close to another irrational number?

I know that there is a rational number arbitrarily close to an irrational, due to the density of real number. But what about an irrational number? Thanks!
1
vote
1answer
45 views

if $|f(x)-f(y)|\le |x-y|^{\sqrt 2}$ then is $f$ a constant function?

if $f: \mathbb{R}\to \mathbb{R}$ satisfies $$|f(x)-f(y)|\le |x-y|^{\sqrt{2}}$$ for all $x,y\in \mathbb{R}$ ,then is f increasing ,decreasing or constant? in my view ,it is clear that $|f(x)-f(y)|$ is ...
4
votes
3answers
73 views

Prove that $\sqrt{6}-\sqrt{2}$ $> 1$.

I'm trying to prove that $\sqrt{6}-\sqrt{2}$ $> 1$. I need to admit that I'm completely new to proof writing and I have completely no experience in answering that kind of questions. However, I came ...
0
votes
2answers
77 views

Let a,b be rationals and x irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$.

I'm trying to solve the following problems: Let $a$,$b$ be rationals and $x$ irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$ Let $x$,$y$ be rationals such that ...
1
vote
1answer
73 views

Is there any attempt to explain irrational numbers from a geometrical point of view?

I'm trying to understand irrational numbers as the result of comparing different referential symmetries, and I'd like to know if there have been any attempt to explain irrationality from any ...
1
vote
1answer
82 views

Is there a fixed integer $n$ for which ${\pi}^{n}$ is prime number?

I would like to know the relationship between $\pi$ and prime numbers distribution ,then I would like to ask if there is a fixed integer for which ${\pi}^{n}$ can be prime or how do i disproof that ...
0
votes
0answers
17 views

Continuity - approximating an irrational number via rationals [duplicate]

If $x=p/q$, where $(p,q)=1$ are integers, then $f(x)=1/q$. If x is irrational then f(x)=0. Prove that: a) f is continuous for all irrationals b) f is not continuous for all rationals. I think ...
0
votes
0answers
26 views

Proving that $z_{k_1,k_2}= k_1r_1+ k_2r_2, k_1, k_2$ is a generator for the real numbers

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by numbers of the form $z_{k_1,k_2}= ...
0
votes
2answers
53 views

Prove that all real numbers can be formed

Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by numbers of the form $z_{k_1,k_2}= ...
5
votes
2answers
451 views

What exactly are those “two irrational numbers” $x$ and $y$ such that $x^y$ is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
2
votes
0answers
66 views

Paper of Paul Erdös

I'm trying to understand On Arithmetical Properties of Lambert Series by Erdös, but am stuck on the first page. He states: Put $k=\left[(\log n)^{1/10}\right]$ and let $p_1,p_2,\ldots$ be the ...