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.
468
questions
787
votes
12
answers
232k
views
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 ...
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]...
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.
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}...
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?
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.
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. $\...
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 ...
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 ...
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?):
$$
\lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases}
1,&x\text{ is rational}\\
0,&x\text{ is irrational}\end{...
30
votes
3
answers
18k
views
Is ArcTan(2) a rational multiple of $\pi$?
Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles
at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$.
Of course the sum of these angles ...
47
votes
1
answer
26k
views
Constructive proof of the irrationality of $\sqrt{2}^{\sqrt{2}}$.
The fact that there exists irrational number $a,b$ such that $a^b$ is rational is proved by the law of excluded middle, but I read somewhere that irrationality of $\sqrt{2}^{\sqrt{2}}$ is proved ...
52
votes
2
answers
9k
views
Why is $\pi$ irrational if it is represented as $C/D$?
$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction.
Then why is $\pi$ an irrational number?
59
votes
5
answers
60k
views
Is there a rational number between any two irrationals?
Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...
21
votes
2
answers
2k
views
How come such different methods result in the same number, $e$?
I guess the proof of the identity
$$
\sum_{n = 0}^{\infty} \frac{1}{n!} \equiv \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x
$$
explains the connection between such different calculations. How ...
19
votes
5
answers
14k
views
For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$ [duplicate]
How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$.
NOTICE that $n$ is in $\mathbb{N}$
Also notice that this is ...
10
votes
4
answers
847
views
Proving that $\sqrt{2}$ is irrational with a math level of a middle school student?
So I have a friend, whose professor challenged the class to prove that $\sqrt{2}$ is irrational by using only middle school math level.
No one managed to do it, and neither did I (although I didn't ...
122
votes
19
answers
13k
views
What is the most unusual proof you know that $\sqrt{2}$ is irrational?
What is the most unusual proof you know that $\sqrt{2}$ is irrational?
Here is my favorite:
Theorem: $\sqrt{2}$ is irrational.
Proof:
$3^2-2\cdot 2^2 = 1$.
(That's it)
That is a ...
24
votes
5
answers
9k
views
Irrationality proofs not by contradiction
Per now, I have basically come upon proofs of the irrationality of $\sqrt{2}$ (and so on) and the proof of the irrationality of $e$. However, both proofs were by contradiction.
When thinking about it,...
15
votes
4
answers
6k
views
Dense set in the unit circle- reference needed
For $x \notin \pi\mathbb Q$, that is, a real $x$ that is not a rational multiple of $\pi$, consider the set $$\{(\cos nx,\sin nx):n = 0,1,2,...\}.$$ It is known that this set is dense in the unit ...
23
votes
6
answers
13k
views
Is $i$ irrational?
On the one hand, $i(=\sqrt{-1})$ cannot be expressed as a ratio of integers, so, by definition, $i$ is not rational $\iff i$ is irrational.
However, the set of irrational numbers, $\mathbb{J}=\mathbb{...
42
votes
2
answers
13k
views
Is sin(x) necessarily irrational where x is rational?
My friend and I were discussing this and we couldn't figure out how to prove it one way or another.
The only rational values I can figure out for $\sin(x)$ (or $\cos(x)$, etc...) come about when $x$ ...
35
votes
1
answer
11k
views
Is there a proof that $\pi \times e$ is irrational?
A little reading suggests:
It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is ...
27
votes
4
answers
37k
views
Sum of two periodic functions is periodic?
I have following paragraph taken from the Stanford's study material.
Question: Is the sum of two periodic functions periodic?
Answer: I guess the answer is no if you are Mathematician, yes ...
5
votes
4
answers
4k
views
Why do irrationality proofs of $\sqrt x$ not apply when $x$ is a perfect square?
When trying to prove that a particular root (say $\sqrt{2}$ or $\sqrt{10}$) cannot be rational, I always see a particular indirect proof that goes something like this:
Suppose $\sqrt{x}$ were ...
3
votes
3
answers
6k
views
Positive integer multiples of an irrational mod 1 are dense [duplicate]
I'm not sure how to solve this one.
Thank you!
$2.$ For any $\alpha\in \mathbb R$ we define $$\lfloor \alpha \rfloor = \max_{n\in\mathbb Z}\{\,n\mid n\leq \alpha\,\}$$ and $$\alpha\bmod 1 = \alpha - \...
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. ...
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
$$\sum\limits_{k=0}^\infty \frac{1}{k!} ...
17
votes
2
answers
3k
views
Why are $e$ and $\pi$ so common as results of seemingly unrelated fields?
I'm sure this gets asked all the time but I swear I googled with no useful result.
What I'm looking for is a reasonably intuitive answer.
Those two constants have some pretty interesting properties. $...
2
votes
7
answers
10k
views
How to prove that $\sqrt 3$ is an irrational number? [duplicate]
Possible Duplicate:
$a^{1/2}$ is either an integer or an irrational number
I know how to prove $\sqrt 2$ is an irrational number. Who can tell me that why $\sqrt 3$ is a an irrational number?
85
votes
1
answer
8k
views
Direct proof that $\pi$ is not constructible
Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass?
Of course, $\pi$ is not constructible because it is transcendental and ...
18
votes
3
answers
2k
views
Closed form for a pair of continued fractions
What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ?
What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ?
It does bear some resemblance to the continued fraction for $e$, which is $2+\cfrac{2}{2+\...
15
votes
5
answers
7k
views
Algorithms for approximating $\sqrt{2}$
Well, "Solving" is the wrong term since I am speaking about irrational numbers. I just don't know which word is the correct word... So that can be part $1$ of my question... what is the correct word ...
13
votes
2
answers
671
views
If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational
Assume there exist some rationals $a, b$ such that $\sqrt[3]{a}, \sqrt[3]{b}$ are irrationals, but:
$$\sqrt[3]{a} + \sqrt[3]{b} = \frac{m}{n}$$
for some integers $m, n$
$$\implies \left(\sqrt[3]{a} ...
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 ...
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:
$$ ...
48
votes
6
answers
85k
views
Proving Irrationality
How is it possible to prove a number is irrational?
First part of that question: How it possible to know that a number will go on infinitely?
Second part: How is it possible to know that no ...
33
votes
12
answers
5k
views
Computing irrational numbers
I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?
22
votes
5
answers
2k
views
Expansion of $(1+\sqrt{2})^n$
I was asked to show that $\forall n\in \mathbb N$ there exist a $p\in \mathbb N^\ast$ such that $$(1+\sqrt{2})^n = \sqrt{p} + \sqrt{p-1}$$
I used induction but it wasn't fruitful, so I tried to use ...
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 ...
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 ...
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'...
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 + ...
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^...
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 ...
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 ...
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 ...
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 ...
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 $n \geq 1$, ...
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 ...