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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1answer
33 views

Methods for Improving Convergence of a sequence of Partial Sums

I have the following sum: $$\zeta(3)+\frac1{4}=\sum_{k=0}^{\infty}\frac{2k^2+7k+7}{(k+1)^3(k+2)(k+3)}$$ Are there any methods that I can use to speed up the convergence of the sequence generated by ...
8
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4answers
517 views

Simplification of an expression involving nested square roots.

I was trying to simplify $\sqrt{14} - \sqrt{16 - 4 \sqrt{7}}$. Numerical evaluation suggested that the answer is $\sqrt{2}$ and it checked out when I substituted $\sqrt{2}$ in the equation $x= ...
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5answers
117 views

Is $\frac{\sqrt7}{\sqrt[3]15}$ rational or irrational.

Is $\frac{\sqrt7}{\sqrt[3]15}$ rational or irrational? Prove it. I am having a hard time with this question. So far what I did was say, assume it's rational, then ...
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1answer
118 views

Show that $\sqrt{2} + \sqrt{3} +\sqrt{5}$ is an irrational number. [closed]

Show that $\sqrt{2} + \sqrt{3} +\sqrt{5}$ is an irrational number.
3
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1answer
48 views

Need a help to show $x<1$

Now I am proving the number $$ \sum_{k=1}^{\infty}\frac{9}{10^{\frac{k(k+1)}{2}}}=0.90900900090... $$ is irrational. Here I use a similar method to the proof of e is irrational by Joseph Fourier. My ...
29
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8answers
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 ...
3
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2answers
61 views

Roots of $z^r=1,r\notin\mathbb{Q}$

If $a,b\in\mathbb{Z}$, and $\frac a b$ is in lowest terms, then $$z^{\frac a b}=1\\\implies z=\exp\left(\frac{2\pi in b}{a}\right)\forall n\in\mathbb{Z}$$ This means that $z$ has exactly $a$ distinct ...
6
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1answer
104 views

Representation of irrationals as $\sum_{n\ge 2}\frac{x_n}{n!}$

Prove that every $x\in(0,1)\setminus\mathbb{Q}$ has a unique representation as $x = \sum_{n\ge 2}\frac{x_n}{n!}$, where $x_n\in\mathbb{Z}_n = \{0,1,2,\ldots,n-1\}$. Probably this is well known, I'd ...
0
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0answers
10 views

Trascendental numbers permutation

Let $x_n$ be the infinite sequence of decimal digits of a fixed irrational/trascendental number. Can I obtain any other irrational/trascendental number's sequence of decimal digits through a ...
5
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3answers
243 views

Proof that the square root of 5 is not a natural number

I am a student and I want to know if this proof is correct. Please help me. Thanks in advance! Proof. If square root of 5 is a natural number, it should be even or odd. If it would be even, we could ...
3
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2answers
105 views

Constructive proof that algebraic numbers form a field

We know that the set $\mathbb{A}$ of algebraic numbers is a field. But there is a constructive proof of this statement? I.e. : given a sum (or a product) of numbers of the form $\sqrt[n]{q}$ with $ q ...
3
votes
1answer
105 views

$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational

If $m_1 , m_2, \cdots m_n$ are natural numbers where at least one of them is not a perfect square, then how do I prove that the sum $$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$$ is irrational? I'm ...
2
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1answer
33 views

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational.

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. Take ...
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1answer
58 views

“Multivariable” version of this lemma about showing analytically that a number is irrational.

Lemma: let $\alpha \in \mathbb{R}^+$ and $p_1,p_2,\dots, q_1, q_2, \ldots \in \mathbb{N}$ such that $\left|\alpha q_n - p_n \right| \neq 0$ for all $n \in \mathbb{N}$ and $$ \lim_{n ...
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4answers
167 views

Is this attempt of showing the square root of two is irrational valid?

Any odd integer squared is always odd, and likewise, any even integer squared is always even. Therefore, the square root of an odd number must be odd and the square root of an even number must be ...
2
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4answers
132 views

Rationality of $e + \pi$

I found just one question similar to this, but it had been edited, so hopefully this isn't asked too often. Given the formulas via infinite sums for expressing $e$ and $\pi$... $$ e = ...
2
votes
1answer
63 views

power e irrational number [closed]

I have a question and it will be appreciated that you tell me some more details. Here is the question. For an arbitrary natural integer n, prove for any $n$, $e^n$ is irrational.
0
votes
1answer
56 views

Is there a systematic method of knowing if $n = \sqrt a $ is or is not a rational number?

Consider $$n = \sqrt a $$ where $a$ is any integer. Is there a rigorous, systematic method of figuring out if $n$ will or will not be a rational number?
1
vote
1answer
44 views

Can we deduce that $⌊r^{n}α⌋≃r^{n}α$ when $r→∞$?

Let $α∈(0,1)$ be an irrational number and let $n≥1$ be a fixed positive integer. For any $r>4$ we define the positive integer $$k=⌊r^{n}α⌋$$ where $⌊.⌋$ denotes the floor function. My question is: ...
0
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2answers
97 views

What do we mean when we say an irrational number can't be expressed as a fraction?

An irrational number is one such that it cannot be expressed by a fraction, but consider the definition of the Golden Ratio. Two line segments, call one a and the other b, are said to be of the ...
8
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2answers
186 views

Proof that at most one of $e\pi$ and $e+\pi$ can be rational

$e$ and $\pi$ are rather peculiar numbers. It turns out that, in addition to being irrational numbers, they are also transcendental numbers. Basically, a number is transcendental if there are no ...
2
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3answers
98 views

Is a prime to the power of a fraction always irrational?

Let $p$ be a prime number and let $x$ be a faction, i.e. $x \in \mathbb{Q} - \mathbb{N}$. It seems to be the case that $p^x$ is always irrational. How do I prove this?
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2answers
305 views

$e^{\pi\sqrt N}$ is very close to an integer for some smallish $N$s. What about $\pi^{e\sqrt N}$?

Heegner numbers (1, 2, 3, 7, 11, 19, 43, 67, 163 - let's use symbol $H_n$) are know for peculiar property that $e^{\pi\sqrt{H_n}}$ are almost integers: $$e^{\pi \sqrt{19}} \approx 96^3+744-0.22$$ ...
0
votes
0answers
36 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
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3answers
42 views

How to evaluate a sum which contains limit variables?

For example: $$\lim_{n\to\infty}\sum_{i=1}^n\frac{n-1}n\frac{1+i(n-1)}n $$ And would the result necessarily be rational, because each term appears to be the multiplication of two rational fractions? ...
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1answer
118 views

Rational summation of irrational numbers

Is the sum of all irrational numbers between any two integer constants rational? I think it should be, because every irrational number should have another irrational with which it would sum to a ...
3
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3answers
479 views

Irrationals forming rationals

Can we obtain every rational number from the multiplication of two irrational numbers? If not, which ones can we not obtain?
3
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2answers
98 views

If $a$ is irrational, does there exist a natural number $n$ such that $na$ is rational?

For some irrational $a$, does there exist an $na$ which is contained within the rational numbers for some natural $n$?
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3answers
488 views

Transcendental a infinitely close to rationals?

Apologies that this question is rather vague, but I do not know how to state it more precisely. Is, say pi, infinitely "close" to some rational number? More importantly, are all transcendental numbers ...
0
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2answers
73 views

Rational and irrational numbers

Consider $x$ a rational number. Let $\epsilon \geq 0$ be the minimal value such that $x + \epsilon$ is irrational, and let also $\gamma > 0$ be the minimal value such that $x+\gamma$ is rational. ...
4
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0answers
78 views

Different types of transcendental numbers based on continued-fraction representation

I've been reading Wikipedia's article on continued fractions. A few examples are given for the continued-fraction representation of irrational numbers: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
1
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1answer
31 views

Does ${\frac{k}{2\left(1-H\right)}} + \frac{1}{H}\in Z$ when $H$ is irrational and $k \in Z^{+}$?

While working on something, I have stumbled across the following expression $$\frac{\Gamma \left({\frac{k}{2\left(1-H\right)}} + \frac{1}{2H}\right)}{\Gamma \left(\frac{1}{2H}\right)}$$ where $0 < ...
1
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1answer
63 views

Find $(1+i)^i$ in simpler terms, without imaginary exponents. [duplicate]

I was asked to find $(1+i)^i$, I don't know what to do when there is an imaginary component in the exponent. since $1+i=\sqrt{2}e^{-\frac{1}{4}i \pi}$ then $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ ...
5
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2answers
349 views

Simple proof that $\pi$ is irrational - using prime factors of denominator

Simple proof that $\pi$ is irrational Consider the Gregory - Leibniz series for $\pi/4$: $$\frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 + \cdots $$ Let $A_n/B_n$ be the irreducible fraction given by ...
2
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3answers
134 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
4
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0answers
72 views

Extention of Euclid's GCD Algorithm. (The Art of Computer Programming, Volume 1, Edition 3, Section 1.2.1, Exercise 12)

Euclid's GCD algorithm which is used to find GCD of two input numbers, say, $c$ and $d$, needs the inputs to be positive integers. Exercise 12 provides an extension to this algorithm and allows $c$ ...
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0answers
34 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
5
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6answers
1k views

Alternate proof for “$\log_{10}{2}$ is irrational”

I need to prove that $\log_{10}{2}$ is irrational. I understand the way this proof was done using contradiction to show that the even LHS does not equal the odd RHS, but I did it a different way and ...
0
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2answers
69 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
2
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4answers
122 views

Prove that $\sqrt{n^2 + 2}$ is irrational

Question: Suppose $n$ is a natural number. Prove that $\sqrt{n^2 + 2}$ is irrational. From looking at the expression, it seems quite obvious to me that $\sqrt{n^2 + 2}$ will be irrational, since ...
0
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1answer
41 views

Sets of irrationals whose square contains a rational

Let $S$ be a subset of the irrationals. Also, lets assume that $S$ has infinitely many elements. My very general question is, under what non-trivial conditions does there exist an element $x\in S$ ...
0
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0answers
114 views

The cube of at least one irrational number is rational

I am supposed to prove the statement above. Here is what I have so far Suppose that the cube of at least one irrational number $n$, is rational. By definition of rational, there exists ...
2
votes
3answers
83 views

Question about $\displaystyle\sum_{n=1}^{\infty}\dfrac{|\sin(n)|}{n}$. [duplicate]

In several places on this site the sum $\displaystyle\sum_{n=1}^{\infty}\dfrac{\sin(n)}{n}$ has been discussed as a generalized alternating series, which therefore converges. I am curious about the ...
0
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1answer
82 views

Real number system

Is the set of rationals a subset of the irrationals? I always assumed it was, but given that irrationals are defined to be numbers that have an infinite, non-repeating decimal expansion, there cannot ...
0
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1answer
87 views

About the continuity of $f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k}$

Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$. Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} ...
0
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1answer
135 views

What's the value of tau?

I've seen $\tau$ on a title of a YouTube video and I need help knowing what the value is. I'm serious. I've never heard of the value. So, what is it? Also, is it rational or irrational (this part ...
2
votes
2answers
55 views

Can $x^{2q}$ be irrational for rational $x$ and $q$?

I think the answer to the question in the title is "yes", because $9^{2/3}$ is irrational by an argument similar to the accepted answer in this question. Or am I mistaken?
6
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3answers
113 views

Irrational numbers in between $n$ and $n+1$

Is the amount of irrationals numbers in between consecutive integers always the same? is this amount infinite?
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2answers
1k views

Proof that $\sin 10^\circ$ is irrational

Today I was thinking about proving this statement, but I really could not come up with an idea at all. I want to prove that $\sin 10^\circ$ is irrational. Any ideas?
0
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1answer
60 views

A positive integer with is not a perfect square is a product of distinct prime factors

This was used as part of the explanation for the following question, but I don't see why it is true. How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect ...