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.

learn more… | top users | synonyms

1
vote
3answers
66 views

Some questions about proofs of irrational numbers

I have some questions about some things I want to clarify in regard to basic questions that ask to show that roots are irrational, for example $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{6}$. To me, I think ...
0
votes
1answer
56 views

About the vertices of a regular polygon in the plane having rational coordinates [on hold]

I have to prove that, except in the case $n=4$, the vertices of a regular $n$-agon in the Euclidean plane cannot have all rational coordinates $(x,y)$. Some idea?
1
vote
1answer
42 views

Randomness in pi and other irrational numbers [duplicate]

This is a post I read about pi while looking for stuff about tau -which is two times as much as pi. This makes me wonder, why does only pi contain such randomness? Don't other non-repeating and ...
1
vote
0answers
173 views

Very simple proof that $\sqrt{2}$ is irrational.

I came across a nice-looking proof that $\sqrt{2}$ is irrational here. It somehow seems to good to be true. What are the assumptions being made in the proof and if this proof is indeed correct, why is ...
2
votes
1answer
38 views

Does a bijection from the reals to the any binary form?

It is fairly simple to store all rational numbers in a binary format (not base 2) (a language composed of only 1s and 0s, no . marking) by simply storing one integer, a seperator, and another integer. ...
-1
votes
1answer
74 views

If $\pi$ is not algebraic number then : is $\pi ^{n}$ algebraic number for $n >1$?

if $\pi$ is not algebraic number then : is $$\pi ^{n}$$ algebraic number for $n >1$ ? Thank you for any kind of help .
2
votes
0answers
46 views

Conway's box function generalized as a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
2
votes
1answer
342 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
5
votes
2answers
71 views

LCM of irrationals

So, I was recently asked by a friend about the lcm of two irrational numbers. As far as I know, mathematically speaking, lcm is generally defined only for positive integers (and sometimes extended to ...
3
votes
0answers
54 views

Transcendence of $\Gamma(1/3), \Gamma(1/4)$

Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a reference to that proof? Or maybe some details on the essential ideas ...
45
votes
13answers
1k 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 ...
3
votes
1answer
254 views

Is a non-zero integral multiple of an irrational number guaranteed to be irrational?

I got to wondering if a non-zero integral multiple of any irrational number is guaranteed to be irrational? This seems intuitive but I can't prove it to myself. There is an answer on this site with ...
0
votes
1answer
38 views

Riemann integral of a non continuous function

We have a function $f : I=[0,1] \rightarrow \mathbb{R}$ defined as: $$f(x)=\begin{cases} 1 &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$ a) Show ...
1
vote
5answers
3k views

plot any irrational number on number line.

I have a basic question that can we plot any irrational number on number line?As I can plot all integers and rational number but how to plot any irrational number on it like $\sqrt2$,$\sqrt3$ etc..
-1
votes
0answers
50 views

Line of Irrational Length? [duplicate]

If we drew a line of irrational length using pythagorean theorem, then is the length of the line really irrational? Can line of irrational length really exist? Will it be possible for a computer with ...
3
votes
0answers
39 views

Do the second-last-digits of the primes $\ge 11$ form a transcendental number?

Suppose, the number $x$ is constructed from the second-last-digits from the primes $\ge 11$ The first $1996$ digits of $x\ =\ 0.11112...$ after the decimal point are : ...
1
vote
1answer
43 views

The irrationality of rapidly converging series

I recently saw a pretty elegant proof of the irrationality of $e$, namely: Let $s_n:=\sum_{k=0}^{n}{\frac{1}{k!}}$ such that $e=\lim_{n\to\infty} s_n$. We obviously have $s_n<e$ and furthermore ...
3
votes
1answer
70 views

Unknown Possible Irrational Number Determination

I have a particular sequence, the coefficients determined by the generating function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty\varepsilon_n\frac{x^n}{n!}$$ The first few numbers are ...
-4
votes
1answer
40 views

The product of xy of two real numbers x and y is irrational then at least one of the x or y must be irrational. [duplicate]

Prove if true or find a counterexample.... The product of $x y $ of two real numbers $x$ and $y$ is irrational then at least one of the $x$ or $y$ must be irrational.
16
votes
3answers
568 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
55 views

why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?

When I was comparing proof for $\sqrt{2}$ and sum of two rational numbers, I found that the proof of two rational number did not mention anything about common factor in the ratio. one proof I found ...
2
votes
0answers
138 views

What is wrong with my reasoning? (Irrational numbers / Fibonacci sequence) [closed]

I just noticed that: We define a procedure $\mathcal{I}:\mathbb{R}\rightarrow \mathbb{R}$ that output a real number. The probability to obtain an irrational or rational number via $\mathcal{I}$ is ...
1
vote
5answers
74 views

Why the set of irrational numbers is represented as $\mathbb{R}\setminus\mathbb{Q}$ instead of $\mathbb{R}-\mathbb{Q}$?

What does the "\" symbol means in this context? I have seen it used for quotient sets like $X /{\sim}$ where $X$ is a set and $\sim$ is an equivalence relation but I don't know what it means applied ...
9
votes
2answers
177 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 ...
0
votes
0answers
58 views

Calculate digits of $\pi$ without needing to reuse them

I am looking for some algorithms that can calculate the digits of $\pi$ without needing to reuse previous digits. I would like to find the most simple and fast algorithms possible. Thanks!
60
votes
1answer
3k 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 ...
311
votes
12answers
88k views

Does Pi contain all possible number combinations?

I came across the following image, which states: $\pi$ Pi Pi is an infinite, nonrepeating (sic) decimal - meaning that every possible number combination exists somewhere in pi. Converted ...
5
votes
4answers
653 views

Number raised to power of irrational number

What is the consequence of raising a number to the power of irrational number? Ex: $2^\pi , 5^\sqrt2$ Does this mathematically makes sense? (Are there any problems in physics world where we ...
0
votes
2answers
56 views

Finding $n$th root of 2 is irrational using given polynomial

The polynomial $f(x)$ is defined by $f(x)=x^n + a_{n-1}x^{n-1}+ \cdots + a_{2}x^2+a_1x+a_0$ where $n \geq 2$ and the coefficients $a_0, \cdots, a_{n-1}$ are integers, with $a_0 \neq 0$. ...
0
votes
0answers
24 views

On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post. I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / ...
28
votes
12answers
2k 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?
3
votes
2answers
95 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
2
votes
1answer
466 views

Irrationality of $\pi$ and circumference to diameter ratio.

How is $\pi$ actually defined? If it is defined as the ratio of the circumference of a circle to its diameter then from this definition itself either of the circumference and diameter has to be ...
16
votes
4answers
237 views

Homework 8th grader: $\pi^2$ is irrational

I'm tutoring a girl in 8th grade (so she is 14 years old) and she recently had a mathematics chapter about numbers. In the last paragraph they introduced the difference between rational and irrational ...
7
votes
2answers
8k views

Is a non-repeating and non-terminating decimal always an irrational?

We can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats). $.0303$ $\cdots$ tends to $\frac{1}{33}$. So,I was wondering this: In the decimal representation, if we start writing the ...
1
vote
2answers
40 views

Show: t(x) = x√2 + √3 is irrational. Hint: consider t(x)²

Earlier in the question we were asked to show that the square root of 6 is irrational, which I did. But I can't seem to figure the last part out. I have included an image for reference. Help is ...
44
votes
9answers
8k views

Is an irrational number odd or even?

My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will ...
2
votes
3answers
85 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?
8
votes
0answers
72 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
-1
votes
2answers
43 views

Does an irrational number $C$ exist such that $C \cdot \sqrt 2 \in \Bbb{Q}$?

Does an irrational number $C$ exist such that $C \cdot \sqrt 2 \in \Bbb{Q}$, where $\sqrt2 \not\mid C$? I just thought of this, I'm trying to find answers that aren't of the form $C=a\sqrt2, ...
-1
votes
4answers
1k views

Prove that $\log_2 3$ is irrational [closed]

Prove that $\log_2 3$ is irrational Seemingly simple homework assignment. Was never the best with logarithms, how would I go about proving?
0
votes
1answer
25 views

Incommensurable units as ratios

I am having a bit of trouble understanding the concept of an incommensurable unit. From what I have gathered so far, it is simply a magnitude that cannot be expressed as the ratio of two natural ...
4
votes
1answer
25 views

Several values of irrational exponentiation

When talking about a number to a rational exponent, there are as many answers as the denominator of the exponent. Like the question: Is $9^{1/2}$ equal to $3$ or $-3$. However when we have an ...
11
votes
1answer
244 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $$S(\alpha)=\big\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \big\}.$$ Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers. ...
2
votes
1answer
36 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
7
votes
0answers
98 views

The sum $\sum_{n=1}^\infty \min_{k\le n}\{\alpha k\}$ for irrational $\alpha$

Let $\alpha$ be an irrational number. For every $n$ let $z_n$ be the integer closest to the number $\alpha n$. Then we can define $$A(\alpha):= \sum_{n=1}^\infty |\alpha n - z_n|.$$ We can also ...
2
votes
2answers
393 views

Proving that any rational number can be represented as the sum of the each cube of three rational numbers

I found the following question in a book: Prove that any integer can be represented as the sum of the each cube of five integers. The answer : ...
9
votes
2answers
139 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)- ...
3
votes
1answer
48 views

Problem understanding this specific proof that $\sqrt{2}$ is irrational.

The proof (taken from http://www.themathpage.com/aPreCalc/rational-irrational-numbers.htm#proof): "To prove that there is no rational number whose square is 2, suppose there were. Then we could ...
17
votes
7answers
8k views

What rational numbers have rational square roots?

All rational numbers have the fraction form $$\frac a b,$$ where a and b are integers($b\neq0$). My question is: for what $a$ and $b$ does the fraction have rational square root? The simple answer ...