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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10
votes
2answers
1k views

Reversing the digits of an infinite decimal

Let $x$ be a real number in $[0,1)$, with decimal expansion $$ x = 0.d_1 d_2 d_3 \cdots d_i \cdots \;. $$ If the decimal expansion is finite, ending at $d_i$, then extend with zeros: $d_k = 0$ for all ...
4
votes
6answers
210 views

How can never ending decimal numbers represent finite lengths? e.g. pi(π), $\sqrt{2}$

Recently, I was in a discussion with a colleague that, whether the πd really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came ...
9
votes
0answers
116 views

Infinitely nested radical expansions for real numbers

Conjecture. For any real number $x \in (0,1]$ there exists a unique expansion in the form $x=-2+\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}$ with $a_k$ being natural numbers from the set $(2,3,4,5,6)$. ...
1
vote
3answers
55 views

Rationalize a surd $\frac{1}{1+\sqrt{2}-\sqrt{3}}$

How can I rationalize the following surd $$\frac{1}{1+\sqrt{2}-\sqrt{3}}$$ What would be the conjugate of the denominator
1
vote
3answers
38 views

exhibit a countable set of irrational numbers with justification

So I have been given this problem and I am totally stumped on what to do...everything I have learned says the irrational numbers are uncountable but I am supposed to exhibit a countable set of ...
1
vote
1answer
41 views

Rationality of $a^2+b^2$

I have looked into this topic lately and have not found an answer to the following question. Is the following true: If $a,b\in\mathbb{R}$ and $a + b$ is rational, then $a^2 + b^2$ is rational
0
votes
1answer
62 views

The irrationality of $\pi/e$ is listed as open yet the infinite product formula for it seems to suggest a way to prove it.

And the formula of all rational products seems to suggest that taking some n as n approaches infinity, the formula will have an always increasing amount of uncancelled primes(so provably non ...
4
votes
2answers
55 views

Rationalising factor of $a+b \sqrt{2}+c \sqrt{3} + d \sqrt{6}$

I am trying to express the inverse of $a+b \sqrt{2}+c \sqrt{3} + d \sqrt{6}$ (given $a, b, c, d \in \mathbb{Q}$) in the form $e+f \sqrt{2}+g\sqrt{3}+h\sqrt{6}$ (where $e, f, g, h \in \mathbb{Q}$). I ...
4
votes
1answer
83 views

How to prove $\log_23$ is irrational?

I think using contradiction is good. Assume $\log_23$ is rational Then $\exists p\in \Bbb{Z}, q\in \Bbb{Z}^*: \log_23 = \frac{p}{q}$ ###$p, q$ has no common factors. Then $3^{q}=2^{p}$ ... Here ...
-1
votes
3answers
62 views

If $x$ is rational and $xy$ is irrational, then $y$ is irrational. [closed]

This is a statement that I need to prove. Let $x$ and $y$ be real numbers. If $x$ is rational and $x\times y$ is irrational, then $y$ is irrational. I believe you have to prove this using ...
-1
votes
2answers
67 views

Suppose that a sequence of rational fractions p/q converge to an irrational number

Suppose that a sequence of (rational) fractions p/q converge to an irrational number r. Show that q converges to infinity.
3
votes
2answers
68 views

Rational Points on $\sin x$ and $\cos x$

Are there any values for $x$ such that both $\sin x$ and $\cos x$ are rational besides $\displaystyle\frac{n\pi}{2}$ and $n\pi$, where $n$ is an integer? I also only want to include $x$ values that ...
1
vote
1answer
37 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
83 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
63 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. ...
1
vote
0answers
38 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
57 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
67 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
112 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
129 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
26 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
133 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
56 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
81 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
90 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
37 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
98 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
200 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
42 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
76 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
69 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
119 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
69 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
134 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 ...
21
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
49 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
63 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 ...