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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382
votes
12answers
105k views

Does Pi contain all possible number combinations?

I came across the following image, which states: $\pi$ Pi Pi is an infinite, nonrepeating (sic) decimal - meaning that every possible number combination exists somewhere in pi. Converted ...
79
votes
4answers
12k views

Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
66
votes
1answer
3k views

Direct proof that $\pi$ is not constructible

Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass? Of course, $\pi$ is not constructible because it is transcendental and ...
64
votes
3answers
695 views

All real numbers in $[0,2]$ can be represented as $\sqrt{2 \pm \sqrt{2 \pm \sqrt{2 \pm \dots}}}$

I would like some reference about this infinitely nested radical expansion for all real numbers between $0$ and $2$. I'll use a shorthand for this expansion, as a string of signs, $+$ or $-$, with ...
56
votes
15answers
2k views

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
47
votes
8answers
3k views

Designing an Irrational Numbers Wall Clock

A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock." There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational. ...
46
votes
9answers
9k views

Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
46
votes
2answers
3k views

Why is $\varphi$ called “the most irrational number”?

I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...
37
votes
4answers
13k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. $\...
32
votes
5answers
22k views

Proof that the irrational numbers are uncountable

Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the ...
32
votes
1answer
829 views

What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions?

I got the number $$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$ in the ...
32
votes
0answers
1k views

Irrationality of $e$

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^...
31
votes
2answers
580 views

Show that $e^{\sqrt 2}$ is irrational

I'm trying to prove that $e^{\sqrt 2}$ is irrational. My approach: $$ e^{\sqrt 2}+e^{-\sqrt 2}=2\sum_{k=0}^{\infty}\frac{2^k}{(2k)!}=:2s $$ Define $s_n:=\sum_{k=0}^{n}\frac{2^k}{(2k)!}$, then: $$ s-...
30
votes
9answers
5k views

Rational + irrational = always irrational?

I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + ...
30
votes
12answers
2k views

Computing irrational numbers

I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?
29
votes
9answers
6k views

How are first digits of $\pi$ found?

Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am ...
29
votes
4answers
4k views

Proving that $m+n\sqrt{2}$ is dense in R

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
29
votes
6answers
29k views

Proving Irrationality

How is it possible to prove a number is irrational? First part of that question: How it possible to know that a number will go on infinitely? Second part: How is it possible to know that no ...
29
votes
4answers
665 views

Linear independence of the numbers $\{1,e,e^2,e^3\}$

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$? The proof should not use that $e$ is transcendental. $e:$ Euler's number. $\{1,e,e^2\}$ is linearly ...
28
votes
10answers
4k views

Is it possible to represent every irrational number as a (limit of) an infinite sum of rational numbers? [duplicate]

For instance, we can certainly represent π in this fashion. $$ \frac{\pi}{4} \;=\; \sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} .\! $$ $\ln(2)$ is also irrational. And even that can be represented as an ...
26
votes
5answers
5k views

How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
24
votes
2answers
2k views

Why is $\pi$ irrational if it is represented as $c/d$?

$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction. Then why is $\pi$ an irrational number?
24
votes
1answer
4k views

Is there a proof that $\pi \times e$ is irrational?

A little reading suggests: It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is ...
24
votes
2answers
876 views

Is there any real number except 1 which is equal to its own irrationality measure?

Is there any real number except $1$ which is equal to its own irrationality measure? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it ...
23
votes
2answers
3k views

Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question. Now, his ...
23
votes
5answers
15k views

Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...
23
votes
0answers
492 views

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

Question : For every even $k\ge 4$, is the following $(\star)$ true? $$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$.}\...
22
votes
4answers
3k 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 ...
22
votes
6answers
5k views

Why must we distinguish between rational and irrational numbers?

The difference between rational and irrational numbers is always stated as: rational numbers can be written as the ratio of two integers, and irrational numbers can't. However, why do mathematicians ...
22
votes
3answers
6k views

irrationality of $\sqrt{2}^{\sqrt{2}}$.

The fact that there exists irrational number $a,b$ such that $a^b$ is rational is proved by the law of excluded middle, but I read somewhere that irrationality of $\sqrt{2}^{\sqrt{2}}$ is proved ...
22
votes
2answers
329 views

Is there an elementary proof that $\sum_{n=1}^\infty {1\over n^s\{n\pi\}}<\infty$ for some $s>0$?

Edit: David Speyer's answer made me realize a couple of things and I would like to clarify. Sorry if the length of this is getting out of hand. First, it is now clear that no estimate can be obtained ...
21
votes
9answers
3k 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 ...
21
votes
3answers
764 views

Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational

Assume we have a sequence of rational numbers $a=(a_n)$. Assume we have a summation function $S: \mathscr {L}^1 \mapsto \mathbb R, \ \ S(a)=\sum a_n$ ($\mathscr {L}^1$ is the sequence space whose sums ...
21
votes
2answers
372 views

Is $\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$ an irrational number?

Obviously: $$\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\cdots=0.1111\dots=\frac{1}{9}$$ is a rational number. Now, if we make terms with demoninators in the form: $$q_n=\sum_{k=0}^{n} 10^k$$ Then ...
21
votes
3answers
2k views

What is the simplest way to prove that the logarithm of any prime is irrational?

What is the simplest way to prove that the logarithm of any prime is irrational? I can get very close with a simple argument: if $p \ne q$ and $\frac{\log{p}}{\log{q}} = \frac{a}{b}$, then because $q^...
21
votes
1answer
523 views

Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient ...
20
votes
6answers
4k views

Prove that $\sqrt 2 +\sqrt 3$ is irrational. [duplicate]

Please prove that $\sqrt 2 + \sqrt 3$ is irrational. One of the proofs I've seen goes: If $\sqrt 2 +\sqrt 3$ is rational, then consider $(\sqrt 3 +\sqrt 2)(\sqrt 3 -\sqrt 2)=1$, which implies ...
20
votes
4answers
2k views

Uncountable set of irrational numbers closed under addition and multiplication?

Is such a thing even possible? There's not much to say really. Obviously if there was a set it would be full of transcendental numbers. This led me to think of a function generating transcendental ...
20
votes
3answers
1k views

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational

Prove $x = \sqrt[100]{\sqrt{3} + \sqrt{2}} + \sqrt[100]{\sqrt{3} - \sqrt{2}}$ is irrational. I can prove that $x$ is irrational by showing that it's a root of a polynomial with integer coefficients ...
20
votes
1answer
651 views

Any proof to $\pi^{e}$'s irrationality?

I've searched for this for a while but get nothing... There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
19
votes
10answers
4k views

Critiques on proof showing $\sqrt{12}$ is irrational.

My only exposure to proofs was in a math logic class I took in University. I was wondering if my attempt at proving that $\sqrt{12}$ is irrational is OK. $$\Big(\frac{m}{n}\Big)^2 = 12$$ $$\Big(\frac{...
19
votes
4answers
1k views

Are there more transcendental numbers or irrational numbers that are not transcendental?

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it unsolvable?...
19
votes
2answers
3k views

Multiples of an irrational number forming a dense subset

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in $[0,1]...
19
votes
1answer
495 views

Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
18
votes
6answers
3k views

Is there a way to write an infinite set that contains only irrational numbers without integer multiples?

Is there a way to write an infinite set that contains only irrational numbers without integer multiples? The infinite set must not contain integer multiples of any other members of that set. For ...
18
votes
6answers
2k views

when product of irrational numbers = rational number?

let $a$ and $b$ be irrational numbers. when do we have $ a \cdot b $ = rational number? for example $\sqrt{2} \cdot \sqrt{2}=2$. I was wondering if there some conditions for the product to be a ...
18
votes
3answers
963 views

Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?

Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}...
18
votes
2answers
510 views

Which results depend on the irrationality of $\pi$?

Recently the following uninteresting clock picture was posted by one of my non-mathematically inclined friends to my facebook wall, saying that it was funny and possibly thinking that I would find it ...
17
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 ...
17
votes
2answers
5k views

ArcTan(2) a rational multiple of $\pi$?

Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$. Of course the sum of these angles ...