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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71 views

Natural bijection between $\mathbb{N}$ and algebraic numbers?

Q. Is there a canonical, explicit bijection between the natural numbers $\mathbb{N}$ and the algebraic numbers? The earlier MSE question, "Bijection for algebraic numbers," does not quite ...
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8answers
231 views

Rational number that approximates $\sqrt{3}$

Questions: Show that is is theoretically possible to find a rational number that approximates $\sqrt{3}$ with an error less than $0.001$. Explain how you would go about determining a ...
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1answer
272 views

For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?

It is well known that there exist two irrational numbers $a$ and $b$ such that $a^b$ is rational. By the way, I've been interested in the following two propositions. Proposition 1 : For each ...
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12answers
5k views

Are there any irrational numbers that have a difference of a rational number?

Are there any irrational numbers that have a difference of a rational number? For example, if you take $\pi - e$, it looks like it will be irrational ($0.423310\ldots$) - however, are there any ...
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2answers
104 views

Can length of line segment have non-terminating decimal form value?

Premise 1: All straight line segments have the value of length equal to the numerical value of the end point, provided the starting point of the line is assigned the numerical value zero. Premise 2: ...
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4answers
1k views

If $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational [closed]

Prove that if $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational. I can understand how it works in my head, I don't know how to prove it though.
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1answer
123 views

On a theorem of Kronecker! [closed]

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: ...
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1answer
63 views

constructing continuous function with range of rational numbers

Can we construct funtion $f$ non-constant and continuous on $\mathbb{R}$, $f$:$\mathbb{R}\rightarrow\mathbb{Q}$. Is that right "By intermediate value theorem, there exist irrational number between ...
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2answers
246 views

Does π start with two identical decimal sequences?

Let d$(x,y)$ be the sequence of decimals of π from the x:th one to the y:th one. My question: is there a number $n$ such that d$(1,n)$ = d$(n+1, 2n)$? I.e., does π start with two identical decimal ...
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2answers
157 views

How can there exist a Liouville number?

I just found out about them today so forgive me if I'm just missing something obvious. As far as I'm aware, the following is true $(\forall a\in\mathbb R\geq0)(((\forall\epsilon\in\mathbb ...
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3answers
1k views

How do I find the value of this weird expression?

How can I find the value of the expression $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^...}} $? I wrote a computer program to calculate the value, and the result comes out to be 2 (more precisely 1.999997). Can ...
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1answer
55 views

Prove that there is an irrational number a such that $2\lt a \lt 3$. Prove also that there is an irrational number $b$ such that $2 \lt b^2 \lt 3$

I tried to prove a is irrational by subtracting two all three sides so I get 0 is less than a-2 less than 1. From there I proved that a-2 is irrational by saying a-2 is equal to rational which ...
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1answer
117 views

Irrational diagonal length problem.

Premise 1: All straight lines have the value of length equal to the numerical value of the end point, provided the starting point of the line is assigned the numerical value zero. Premise 2: No ...
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3answers
93 views

How to simply this fraction with irrational denominators? [closed]

How to simplify? $\frac{1}{1+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{7}} \frac{1}{\sqrt{7}+3}$
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0answers
60 views

Prove that $E_0$ is transcendental

Consider the non-negative natural numbers: $0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19\dots$ Encode the primes as $1$, the rest as $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ ...
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2answers
63 views

Seeing the plane as a four (or more) dimensional vector space on $\mathbb Q$

As I was trying to answer a question about the enumeration of circuits one can build with a set of miniature train track elements, I realized that all plane positions that could be reached had ...
4
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4answers
209 views

How to compute a lot of digits of $\sqrt{2}$ manually and quickly?

After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in ...
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0answers
90 views

What is the name of this irrational math constant and is there a compact way to write it? 0.10110111011110…

I think this number is a transcendental number and I've tried looking online to see who first made it, I'm not sure if it's a Liouville Number or if there is a more common or better name for it. Does ...
2
votes
1answer
48 views

Algorithm for eliminating irrationality in denominator

Good day. Suppose $a$ is rational number, $p$ is positive integer and $a^{1/p}$ is irrational. If we want to eliminate irrationality in the denominator of the fraction $\frac{1}{a^{1/p}}$, then there ...
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5answers
8k views

why is PI considered irrational if it can be expressed as ratio of circumference to diameter? [duplicate]

Pi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where both ...
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1answer
43 views

Rational Number Density in a Square

It is well known that rational numbers are distributed on the number line everywhere compactly. If we consider a 'square' a parallelogram to be precise, formed by natural numbers p and q, i.e. ...
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1answer
41 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 ...
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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 ...
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2answers
70 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 ...
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1answer
109 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 ...
3
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1answer
81 views

Can permutating the digits of an irrational/transcendental number give any other such number?

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 ...
3
votes
2answers
126 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
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1answer
115 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 ...
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1answer
107 views

Prove that the quotient of a nonzero rational number and an irrational number 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 $m,n\in \mathbb Z, n\neq 0$. ...
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2answers
81 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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1answer
47 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: ...
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2answers
129 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 ...
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2answers
359 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: $$e^{\pi \sqrt{19}} \approx 96^3+744-0.22$$ ...
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1answer
249 views

Is the sum of all irrational numbers between two integers rational?

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 ...
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3answers
486 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?
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3answers
502 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 ...
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2answers
99 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. ...
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0answers
111 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: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
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1answer
34 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 < ...
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1answer
73 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 $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ ...
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0answers
92 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.
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2answers
150 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?
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1answer
52 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 $x\in S$ ...
2
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3answers
94 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 ...
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1answer
163 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 ...
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1answer
149 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} ...
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1answer
458 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 ...
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3answers
121 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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1answer
128 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 ...