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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2
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1answer
200 views

Rational Irrational Numbers

I know that a rational number can always be expressed as a fraction, but can't we also say that it is a number that follows a definite pattern? Like one-third for example; it is never ending as a ...
8
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2answers
156 views

when index is irrational number with inequality

Let $x>0$, show that $$x^{\sqrt{3}}+x^{\frac{\sqrt{3}}{2}}+1\ge 3\left(\dfrac{1+x}{2}\right)^{\sqrt{3}}$$ we consider $$f(x)=2^{\sqrt{3}}(x^{\sqrt{3}}+x^{\dfrac{\sqrt{3}}{2}}+1)- ...
10
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1answer
109 views

If all the numbers $(1^\alpha,\,2^\alpha,\,3^\alpha,\,\dotsc)$ are integer, then $\alpha$ is an integer.

A theorem of Siegel asserts that If $\beta>0$ and $2^\beta,\,3^\beta,\,5^\beta$ are integers, then $\beta$ is an integer. The following result is a beautiful consequence of this theorem ...
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5answers
54 views

Irrational number multiplied by its fractional part becomes rational (SOLVED)

Here's a Korean middle school midterm problem I've been struggling for quite some time now. "$X$ is an irrational number such that $X>0$, and $Y$ is fractional part of $X$. If $$X^2+Y^2=27$$, find ...
8
votes
3answers
202 views

Irrational Numbers : Show that $0.1248163264…$ is irrational

I was working through some basic Number Theory Problems in Rosen and came across the following problem : Show that the real number $0.1248163264...$ represented in ...
8
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3answers
528 views

“Length” of rationals in an interval

For $x \in \mathbb{R}$, define $r(x)$ as follows: $$ r(x)= \begin{cases} 1 &\text{if $x$ is rational},\\ 0 &\text{if $x$ is irrational}. \end{cases} $$ Q. What is $\int_0^1 r(x) dx$ ? I ...
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2answers
87 views

Difficult Complex Number Proof. Given $|w| =1$ or $|v|=1$ [closed]

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\left|\frac{w-z}{1-\overline{w}z}\right| = 1$$ Hint: Note that $|a|^2 = a\overline a$ I have been ...
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4answers
93 views

Definition of irrational number

What is a formal definition of a irrational number? Usually, we say that it is a number that it is not rational. Is it enough?
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2answers
79 views

Closeness of $n! \ x$ to integers for irrational $x$

This question came up in the comments to another question. Is there an irrational number $x$ such that, for sufficiently large $n$, the product $$ n! \ x $$ is arbitrarily close to an integer? More ...
3
votes
2answers
49 views

When do we know for sure that we have the correct digits of an irrational number?

This comes from a programming assignment I was given using MATLAB. The objective was to calculate the difference between $\pi/4$ and the Leibniz series for computing $\pi/4$ with $n = 200$. This ...
2
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0answers
36 views

Continued fraction for $[1,2,3,4,5,6,\dots]$ [duplicate]

Any continued fraction that does not terminate or repeat can't be rational or a quadratic irrational. It is not hard to write something that does not fit these two categories. Can we still get a ...
0
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1answer
77 views

Show that $4^\frac{1}{3}$ is an algebraic number?

How do you show that $4^\frac{1}{3}$ is an algebraic number? I don't understand the question nor how to begin on describing the proof to show what the question is asking.
1
vote
0answers
74 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 ...
1
vote
8answers
234 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 ...
8
votes
1answer
276 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 ...
8
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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 ...
1
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2answers
107 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: ...
3
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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.
0
votes
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: ...
0
votes
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 ...
10
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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 ...
3
votes
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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votes
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 ...
0
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1answer
56 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
122 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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votes
3answers
95 views

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

How to simplify? $\frac{1}{1+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{7}} \frac{1}{\sqrt{7}+3}$
3
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0answers
61 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 $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ ...
2
votes
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
votes
4answers
210 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 ...
2
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0answers
92 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 ...
5
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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 ...
0
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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. ...
1
vote
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 ...
29
votes
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 ...
3
votes
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 ...
6
votes
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
votes
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
131 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
117 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
119 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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vote
1answer
48 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
130 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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votes
2answers
365 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
258 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?
8
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3answers
505 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
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
114 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: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...