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
votes
1answer
45 views

What is the rate of decay of $\min\{k\xi-\lfloor k\xi\rfloor|k\in\{1,\dots,n\}\}$, for irrational $\xi$?

I wish to establish bounds on the sequence of infima of $\{n\xi\}_{n\in\Bbb N}$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part function and $\xi$ is irrational. I can prove that ...
10
votes
4answers
593 views

How to show $\sqrt{4+2\sqrt{3}}-\sqrt{3} = 1$

I start with $x=\sqrt{4+2\sqrt{3}}-\sqrt{3}$, then $\begin{align*} x +\sqrt{3} &= \sqrt{4+2\sqrt{3}}\\ (x +\sqrt{3})^2 &= (\sqrt{4+2\sqrt{3}})^2\\ x^2 + (2\sqrt{3})x + 3 &= 4+ 2\sqrt{3}\\ ...
4
votes
5answers
757 views

How to know a irrationals never repeat?

How would you respond to a middle school student that says: “How do they know that irrational numbers NEVER repeat? I mean, there are only 10 possible digits, so they must eventually start repeating. ...
0
votes
2answers
52 views

Boundaries for Specific Sets with Ambient Space $\mathbb{R}$

I'm trying to find the boundaries for each the following sets: (a) $\begin{Bmatrix}\frac{1}{n}:n\in\mathbb{N}\end{Bmatrix}\overset{?}{=}\{1\}$ (b) $[0,3]\cup(3,5)\overset{?}{=}\{0,5\}$ (c) ...
6
votes
2answers
736 views

Sum of two irrational radicals is irrational?

If $a,b,m$ and $n$ are positive integers such that $\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational numbers, how can we prove that the sum $\sqrt[m]{a}+\sqrt[n]{b}$ is also irrational?
11
votes
1answer
384 views

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

Does someone know a proof that $\{1,\pi,{\pi}^2\}$ is linearly independent over $\mathbb{Q}$ ? The proof should not use that $\pi$ is transcendental. $\{1,e,e^2,e^3\}$ is linearly independent over ...
9
votes
1answer
596 views

Proving the irrationality of $e^n$.

Let $n$ be a positive integer. I know the traditional proof that $e$ is irrational. How do we show that $e^n$ is irrational in some sort of similar line? I am of course assuming it is but I would be ...
27
votes
4answers
628 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 ...
-3
votes
1answer
197 views

The Irrationality of 2

I am sorry it is not 'research level'. A quick answer will do. When I attempt using the Square root of 2 method to prove the rationality of Square root of 4 according to how it was done in a book, 2 ...
1
vote
2answers
76 views

Integer outputs of $y=x^2$ , do their last digits form an irrational?

Let the domain of $y=x^2$ be the positive integers. I input consecutive positive integers from $[1, \infty)$ their last digits are $a, b, c, ...$ respectively. If I then make the number $z=\frac ...
3
votes
1answer
115 views

Is there any kind of irrational number wich does not contain digit 9?

At first we must prove that there is or is`t irrational numbers which does not contain digit 9! if there are many kind of such numbers, then there is another question: how to write down algebraic ...
5
votes
3answers
609 views

property of real number system

"Between every two rational numbers there exist infinite irrational numbers and between every two irrational numbers there exist infinite rational numbers. Is this statement correct? If it is, then ...
0
votes
3answers
337 views

how to find out any digit of any irrational number?

We know that irrational number has not periodic digits of finite number as rational number. All this means that we can find out which digit exist in any position of rational number. But what about ...
4
votes
2answers
119 views

What irrational number has the simplest calculation in terms of computation?

I came across https://github.com/philipl/pifs which is a fancy way of storing data. And a thought struck my mind, is it so that Pi is the simplest irrational number to calculate? So the Question is. ...
1
vote
1answer
186 views

how do we know the BBP formula for $\pi$ is valid?

I recently read about the Bailey–Borwein–Plouffe formula for calculating the $n^{\rm th}$ digit of $\pi$. I'm curious to how can we be sure that the formula is always accurate or correct?! Even if we ...
4
votes
0answers
181 views

Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books , articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.
2
votes
1answer
95 views

Curious function problem (EDIT: Not so curious, but didn`t see it at the time of writing)

This one is directly from my head and although it could be something trivial I do not see the way to attack it but the problem looks interesting and I want to share it with you, here it is: Let us ...
1
vote
0answers
119 views

Gelfond-Schneider Constant $2^{\sqrt{2}}$

Someone knows a proof (books , articles) that $2^{\sqrt{2}}$ is irrational ? Without using that $2^{\sqrt{2}}$ is transcendent. Any hints would be appreciated.
3
votes
2answers
609 views

Find the limiting value of the sequence

A sequence is given by the recurrence relation: $$u_n = 1 + {1\over u_{n-1} +1}, u_1 = 1, n{=\ge}1$$ Work out the 2nd, 3rd and 4th term of the sequence and find the limiting value of the sequence. ...
4
votes
3answers
152 views

prove that $\sqrt{2}$ is irrational using only geometry

Prove that $\sqrt{2}$ is irrational using only geometric concepts and proofs. The proof should look like a proof in Euclid's elements or standard high school geometry. No algebra is allowed. (I know ...
1
vote
0answers
105 views

Continued fraction of $\gamma+1$ using recursion

Number $\gamma,$ the Euler-Mascheroni constant, is defined as the value of $$\gamma = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k} - \ln(n).$$ We know that $$\lim_{n\to\infty} ...
1
vote
2answers
91 views

Irrationality proof by rational approximations

Assume we have a sequence of rational numbers $\left(\frac{p_n}{q_n}\right),$ where $\gcd(p_n,q_n)=1, \ \forall n \in \mathbb N$. We know that $$\lim_{n\to\infty} \left(\frac{p_n}{q_n}\right)= x$$ ...
4
votes
1answer
80 views

A further question on the irrationality of $x^2+y^2=3$

(Apologies for a further question on the same problem) On page 79 of Julian Harvil's book "The Irrationals" he sets out to prove (by contradiction) that all the points on the circle described by ...
5
votes
3answers
144 views

Irrationality of the points on $x^2 + y^2 = 3$

On page 79 of his book "The Irrationals", Julian Harvil sets out to prove that all the points on the Cartesian plane of the circle described by $x^2 +y^2 =3$ are irrational... (paraphrased below) ...
0
votes
2answers
94 views

$\pi$ does not lie in any quadratic extension of $\mathbb{Q}$

Knowing that $\pi^2$ is irrational: How can we prove that $\pi$ does not lie in any quadratic extension of $\mathbb{Q}$ ? Without using that $\pi$ is transcendent. Any hints would be appreciated.
14
votes
3answers
638 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? I doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = ...
1
vote
1answer
593 views

Prove $a + b\sqrt{2}$ is irrational

Suppose that a and b are non-zero rational numbers. How can I show that $a+b√2$ is not a rational number. You may assume that $√2$ is not a rational number. I thought that finding contradictions in ...
5
votes
1answer
152 views

If $q>1$ is not an integer, can $q^n$ be made arbitrarily close to integers?

This question arose when I heard about Mill's constant: the number $A$ such that $\lfloor A^{3^n} \rfloor$ is prime for all $n$. It made me wonder whether $A^{3^n}$ could be made arbitrarily close to ...
0
votes
1answer
94 views

Transcendental proofs vs. Irrational proofs

Why are proofs of the transcendence of certain numbers usually harder than irrationality proofs of those same numbers (for example, Lindemann's proof of the transcendence of pi vs. Niven's proof of ...
2
votes
3answers
939 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 ...
22
votes
2answers
2k 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 ...
1
vote
1answer
56 views

How do you show this isomorphism?

How do you show that $\mathbb{R} \backslash \mathbb{Q} \cong \mathbb{N}^{\mathbb{N}}$? What is a good starting point in showing this?
17
votes
3answers
5k 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 ...
0
votes
0answers
58 views

Any irrational number can be raised to a power so that the result is an integer number [duplicate]

Does it hold in general, that for every irrational number there exists a power to which when raised, the result will be an integer? Does there exist a counterexample, of which it can be showed that no ...
7
votes
4answers
2k views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
3
votes
4answers
237 views

What's $P$ and what's $Q$ in this classic proof of the irrationality of $\sqrt 2$?

In this proof extracted from the Wikipedia A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. If it were rational, it could be expressed as ...
0
votes
1answer
42 views

How to solve the following problems with exponent?

If $9^{x+2}= 240+9^x$ then x= ? $10^x = 64$ what is the value of $10^{(x/2)+1} = ?$ $x/x^{1.5} = 8*x^{-1}$ and x > 0 , then x = ? $x^{-2} = 64$, then $x^{1/3} + x^0$ = ? $4^x - 4^{x-1} = 24 $ then ...
0
votes
2answers
3k views

Numbers that cannot be expressed as fractions

What are Numbers that cannot be expressed as Fractions called?
2
votes
1answer
65 views

Irrational sum to integers?

Is it possible for $(a-b)k + bf$ to be an integer if $k,f$ are irrational numbers and $a,b$ are integers? What about $(a-b)k - bf$?
4
votes
0answers
148 views

Can every string of numbers be found in the number pi (cfr. infinite monkey theorem)? [duplicate]

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of ...
6
votes
3answers
698 views

Is $e^{i+\pi}$ irrational or not?

Since we know that the value of $e$, $i$, and $\pi$ are irrational reals, how about $$e^{i+\pi}\;?$$ Is it still irrational (that is, not a Gaussian rational)? The problem make me curious until now.
16
votes
2answers
477 views

What is the value of $\sum\limits_{i=1}^\infty\frac{1}{p_{p_i}}$ where $p_{i}$ is the $i$th prime?

What is the value of $\sum\limits_{i=1}^\infty\dfrac{1}{p_{p_i}}$ where $p_n$ is the nth prime (and so $p_{p_n}$ is the $k$th prime, where $k$ is the $n$th prime) ? Thus ...
12
votes
7answers
495 views

Numbers with no finite representation on paper

It occurred to me that there must be a lot of numbers without any form of finite representation on paper. Is there a name for these numbers? For example... Integers and rationals have a very simple ...
3
votes
3answers
143 views

More three-term arithmetic progression questions

This question was inspired by a recent question about whether $\frac1{2}$, $\frac1{3}$ and $\frac1{15}$ can be (possibly non-consecutive) terms in an arithmetic progression. My question(s): Which of ...
2
votes
1answer
265 views

Geometric proof that if n is a non-perfect square, then √n is irrational.

I know there is a geometric proof of the irrationality of √2. I thought maybe this one could be generalized for √n when n is a non-perfect square, but I could not find something like that anywhere. ...
2
votes
2answers
89 views

Could you prove convergence?

Two questions: Prove $a_n$ is bounded by 2 if $a_1=0, a_2=\sqrt2,\ldots,a_{n+1}=\sqrt{2+a_n}$ Prove (I've already proven $s_n$ is monotonically increasing and bounded above by 3) $\lim_{n \to ...
16
votes
3answers
583 views

Is there a simple proof that $e^2$ is irrational using a positional numeral system?

My favorite proof that $e$ is irrational goes something like this. Observe that we can write any real number $r$ as $$ a \,+\, \frac{b_2}{2} \,+\, \frac{b_3}{3!} \,+\, \frac{b_4}{4!} \,+\, ...
0
votes
1answer
687 views

Check my answer: $\Bbb Q$ is neither open nor closed

I have a so easy question. I have done it's answer by myself. I want you to only check my answer please. Does there exist any mistake or the missing? The set of irrational numbers - $\Bbb Q$ is ...
16
votes
4answers
9k 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 ...