Questions tagged [irrational-numbers]

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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787 votes
12 answers
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Does $\pi$ contain all possible number combinations?

$\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of ...
Chani's user avatar
  • 7,871
47 votes
3 answers
13k 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]...
squiggles's user avatar
  • 1,903
66 votes
6 answers
23k views

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

I am having trouble proving the statement: Let $$S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that for every $\epsilon > 0$, the intersection of $S$ and $(0, \epsilon)$ is nonempty.
user11135's user avatar
  • 773
42 votes
5 answers
7k views

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

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}...
Spine Feast's user avatar
  • 4,760
130 votes
7 answers
41k 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?
John Hoffman's user avatar
  • 2,734
25 votes
3 answers
10k views

For an irrational number $\alpha$, prove that the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$ [closed]

I am not able to prove that this set is dense in $\mathbb{R}$. Will be pleased if you help in a easiest way. $A=\{a+b\alpha: a,b\in \mathbb{Z}\}$ where $\alpha$ is a fixed irrational number.
Myshkin's user avatar
  • 35.9k
161 votes
5 answers
62k 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. $\...
tel's user avatar
  • 1,863
69 votes
4 answers
80k 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 ...
nkassis's user avatar
  • 841
21 votes
5 answers
6k views

For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense 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 ...
EQJ's user avatar
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26 votes
3 answers
8k views

Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals

I stumbled upon this "relation" (is the name correct?): $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is irrational}\end{...
rubik's user avatar
  • 9,344
30 votes
3 answers
18k views

Is 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 ...
Joseph O'Rourke's user avatar
47 votes
1 answer
26k views

Constructive proof of the 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 ...
user avatar
52 votes
2 answers
9k 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?
Sam's user avatar
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59 votes
5 answers
60k 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 ...
MJD's user avatar
  • 65.4k
21 votes
2 answers
2k views

How come such different methods result in the same number, $e$?

I guess the proof of the identity $$ \sum_{n = 0}^{\infty} \frac{1}{n!} \equiv \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x $$ explains the connection between such different calculations. How ...
Luke's user avatar
  • 2,328
19 votes
5 answers
14k views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$ [duplicate]

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
Guest_000's user avatar
  • 809
10 votes
4 answers
847 views

Proving that $\sqrt{2}$ is irrational with a math level of a middle school student?

So I have a friend, whose professor challenged the class to prove that $\sqrt{2}$ is irrational by using only middle school math level. No one managed to do it, and neither did I (although I didn't ...
John Mayne's user avatar
  • 2,148
122 votes
19 answers
13k 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 ...
marty cohen's user avatar
24 votes
5 answers
9k views

Irrationality proofs not by contradiction

Per now, I have basically come upon proofs of the irrationality of $\sqrt{2}$ (and so on) and the proof of the irrationality of $e$. However, both proofs were by contradiction. When thinking about it,...
Fredrik Meyer's user avatar
15 votes
4 answers
6k views

Dense set in the unit circle- reference needed

For $x \notin \pi\mathbb Q$, that is, a real $x$ that is not a rational multiple of $\pi$, consider the set $$\{(\cos nx,\sin nx):n = 0,1,2,...\}.$$ It is known that this set is dense in the unit ...
martin's user avatar
  • 931
23 votes
6 answers
13k 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, $\mathbb{J}=\mathbb{...
beep-boop's user avatar
  • 11.6k
42 votes
2 answers
13k views

Is sin(x) necessarily irrational where x is rational?

My friend and I were discussing this and we couldn't figure out how to prove it one way or another. The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
Queequeg's user avatar
  • 884
35 votes
1 answer
11k 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 ...
idmercer's user avatar
  • 2,521
27 votes
4 answers
37k views

Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
Bibek Subedi's user avatar
5 votes
4 answers
4k views

Why do irrationality proofs of $\sqrt x$ not apply when $x$ is a perfect square?

When trying to prove that a particular root (say $\sqrt{2}$ or $\sqrt{10}$) cannot be rational, I always see a particular indirect proof that goes something like this: Suppose $\sqrt{x}$ were ...
KutuluMike's user avatar
3 votes
3 answers
6k views

Positive integer multiples of an irrational mod 1 are dense [duplicate]

I'm not sure how to solve this one. Thank you! $2.$ For any $\alpha\in \mathbb R$ we define $$\lfloor \alpha \rfloor = \max_{n\in\mathbb Z}\{\,n\mid n\leq \alpha\,\}$$ and $$\alpha\bmod 1 = \alpha - \...
InfimumMaximum's user avatar
20 votes
9 answers
69k views

Prove that the product of a non-zero rational and irrational number is irrational.

Could you please confirm if this proof is correct? Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational. Proof: Proof by contradiction, we assume that $qy$ is rational. ...
persepolis's user avatar
17 votes
5 answers
13k views

e is irrational

Prove that e is an irrational number. Recall that $\,\mathrm{e}=\displaystyle\sum_{n=0}^\infty\frac{1}{n!},\,\,$ and assume $\,\mathrm{e}\,$ is rational, then $$\sum\limits_{k=0}^\infty \frac{1}{k!} ...
terrible at math's user avatar
17 votes
2 answers
3k views

Why are $e$ and $\pi$ so common as results of seemingly unrelated fields?

I'm sure this gets asked all the time but I swear I googled with no useful result. What I'm looking for is a reasonably intuitive answer. Those two constants have some pretty interesting properties. $...
Luka Horvat's user avatar
  • 2,618
2 votes
7 answers
10k views

How to prove that $\sqrt 3$ is an irrational number? [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number I know how to prove $\sqrt 2$ is an irrational number. Who can tell me that why $\sqrt 3$ is a an irrational number?
Chen's user avatar
  • 91
85 votes
1 answer
8k 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 ...
lhf's user avatar
  • 216k
18 votes
3 answers
2k views

Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is $2+\cfrac{2}{2+\...
Angela Pretorius's user avatar
15 votes
5 answers
7k views

Algorithms for approximating $\sqrt{2}$

Well, "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be part $1$ of my question... what is the correct word ...
Albert Renshaw's user avatar
13 votes
2 answers
671 views

If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational

Assume there exist some rationals $a, b$ such that $\sqrt[3]{a}, \sqrt[3]{b}$ are irrationals, but: $$\sqrt[3]{a} + \sqrt[3]{b} = \frac{m}{n}$$ for some integers $m, n$ $$\implies \left(\sqrt[3]{a} ...
Gerard's user avatar
  • 4,264
7 votes
3 answers
1k views

Is this proof that $\sqrt 2$ is irrational correct?

Suppose $\sqrt 2$ were rational. Then we would have integers $a$ and $b$ with $\sqrt 2 = \frac ab$ and $a$ and $b$ relatively prime. Since $\gcd(a,b)=1$, we have $\gcd(a^2, b^2)=1$, and the fraction ...
MJD's user avatar
  • 65.4k
4 votes
2 answers
596 views

How many digits of accuracy will an answer have?

I was doing a project Euler problem where I needed to find several Fibonacci numbers, but their index was so large that I could not use the typical recursive method. Instead, I used Binet's rule: $$ ...
Ryan's user avatar
  • 1,200
48 votes
6 answers
85k 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 ...
Peter's user avatar
  • 1,021
33 votes
12 answers
5k 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?
user avatar
22 votes
5 answers
2k views

Expansion of $(1+\sqrt{2})^n$

I was asked to show that $\forall n\in \mathbb N$ there exist a $p\in \mathbb N^\ast$ such that $$(1+\sqrt{2})^n = \sqrt{p} + \sqrt{p-1}$$ I used induction but it wasn't fruitful, so I tried to use ...
Meadara's user avatar
  • 601
7 votes
1 answer
271 views

Normal Numbers as members of a larger set?

Is there a named set of numbers (containing, as a subset, the Normal Numbers) comprised of numbers that contain every finite sequence of digits at least once? (The Normal Numbers have all finite ...
Steve Lord's user avatar
6 votes
1 answer
1k views

Proof that $\sqrt[m]{a} + \sqrt[n]{b}$ is irrational

Is there a way to prove that $\sqrt[m]{a} + \sqrt[n]{b}$ ($\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational); $a, b, m, n \in \mathbb{N}$; $m, n \neq 2$; is irrational without using the theorem mentioned ...
ArthurN's user avatar
  • 63
1 vote
2 answers
2k 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 I'...
Katlyn Edwards's user avatar
43 votes
11 answers
31k 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 + ...
gator's user avatar
  • 1,835
26 votes
3 answers
4k 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^...
Dan Brumleve's user avatar
  • 17.8k
19 votes
3 answers
5k views

Real Numbers to Irrational Powers

In a related question we discussed raising numbers to powers. I am interested if anybody knows any results for raising numbers to irrational powers. For instance, we can easily show that there ...
Joshua Shane Liberman's user avatar
16 votes
2 answers
4k views

Is there a proof that $\pi$ is an irrational number?

Most math texts claim that $\pi$ is an irrational number. However, I'm having a little bit of trouble understanding that. Since nobody has calculated all of the digits of $\pi$, how can we know that ...
Nathan Osman's user avatar
  • 1,873
10 votes
3 answers
712 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 ...
M_F's user avatar
  • 327
9 votes
3 answers
829 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 ...
XPenguen's user avatar
  • 2,281
8 votes
3 answers
3k views

Convergence properties of $z^{z^{z^{...}}}$ and is it "chaotic"

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
cpiegore's user avatar
  • 1,490
8 votes
2 answers
2k views

rational angles with sines expressible with radicals [duplicate]

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...
Angela Pretorius's user avatar

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