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
8 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
90 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
45 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 ...
2
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3answers
73 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 ...
4
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2answers
77 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
33 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
27 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
60 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
82 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
31 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
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2answers
188 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
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2answers
29 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
73 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
46 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
45 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 ...
-10
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0answers
93 views

Is the Riemann Hypothesis incorrect? [closed]

See the attached image I would like to know your opinion about if the zeros shown in this picture can be considered as the zeros mentioned by Riemann in his Z function. I think yes and that his ...
0
votes
2answers
38 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 ...
0
votes
3answers
60 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
90 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
90 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
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3answers
65 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
85 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 ...
19
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
21 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 ...
0
votes
2answers
34 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
57 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
157 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
32 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
40 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
72 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
65 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
71 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
73 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
11 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 I ...
0
votes
0answers
25 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
51 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
362 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
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0answers
65 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 ...
2
votes
1answer
89 views

Show $e^x$ is irrational for rational $x \neq 0$ [duplicate]

I want to show that if $x$ is rational and nonzero then $e^x$ is irrational. Clearly $e^{\frac{r}{s}} = \frac{p}{q} \Rightarrow q^s e^r = p^s$, but this doesn't seem helpful. The usual proof that $e$ ...
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7answers
1k views

How can a Cauchy sequence converge to an irrational number?

I am a physics major and would like to clear a confusion regarding complete metric spaces. I am quoting the definition of a Cauchy sequence from wikipedia below Formally, given a metric space $(X, ...
1
vote
0answers
25 views

the summation goes to the integration

Let $f$ be a continuous function on $\mathbb R$ with period $1$. Show that for any irrational number $x$, $\frac { \sum_{k =1 }^ N f (k x ) } { N} \to \int _ 0 ^1 f(t) \, d t $ as $N \to \infty$. I ...
1
vote
0answers
35 views

How do I know if this will be rational or irrational? ($a^b$)

Usually, when I have $a^b$ when $a$ and $b$ are both irrational, I assume that it will be irrational. But that is not always true, I assume, so when is the result irrational? How will I know? Take ...
11
votes
1answer
219 views

Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational?

Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational, where $p_n$ is the $n^{\text{th}}$ prime number? This question is spurred by the comment thread on this question where I presented a rough idea of a ...
4
votes
4answers
134 views

Why are there more Irrationals than Rationals given the density of $Q$ in $R$?

I'm reading "Understanding Analysis" by Abbott, and I'm confused about the density of $Q$ in $R$ and how that ties to the cardinality of rational vs irrational numbers. First, on page 20, Theorem ...
5
votes
3answers
217 views

$\sqrt[31]{12} +\sqrt[12]{31}$ is irrational

Prove that $\sqrt[31]{12} +\sqrt[12]{31}$ is irrational. I would assume that $\sqrt[31]{12} +\sqrt[12]{31}$ is rational and try to find a contradiction. However, I don't know where to start. Can ...
3
votes
1answer
92 views

Is $\frac{1}{2^{2^{0}}}+\frac{1}{2^{2^{1}}}+\frac{1}{2^{2^{2}}}+\frac{1}{2^{2^{3}}}+…$ algebraic or transcendental?

Inspired by this question, the series $\dfrac{1}{2^{2^{0}}}+\dfrac{1}{2^{2^{1}}}+\dfrac{1}{2^{2^{2}}}+\dfrac{1}{2^{2^{3}}}+\dots$ is clearly irrational. But is it algebraic or transcendental? I ...
0
votes
0answers
83 views

How to prove $e^{n}$ is irrational? [duplicate]

How to prove $e^{n}$ is irrational for $n=2,3,4,5,...$ where $e$ is natural logarithm constant?
2
votes
3answers
53 views

How is this limit being solved? I can't grasp it

I am going over limits for my finals as I notice this example in my schoolbook discribing limits of the undefined form $0\over0$ in the shape of an irrational fraction. $$\lim\limits_{x \to 1} ...
2
votes
2answers
170 views

Approximation of $\pi$ using Brahmagupta's Identity

Brahmagupta, an ancient Indian Mathematician, gave an pretty efficient algorithm for finding integer solutions to the famous Pell's Equation, far before Fermat propounded this before the European ...
5
votes
1answer
59 views

Can finite sums of two numbers come arbitrarily close to zero?

Given two real numbers $a$ and $b$, define an $a$-$b$-sum as a finite sum of $a$'s and $b$'s, i.e. a sum: $$m\cdot a + n\cdot b$$ where $m,n$ are non-negative integers. Is there a pair of numbers ...