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
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1answer
29 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 ...
2
votes
1answer
37 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)| ...
5
votes
4answers
674 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 ...
3
votes
1answer
50 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 ...
1
vote
2answers
35 views

Find the relationship between $n$ and $m$ (both natural numbers) such that $m^{1/n}$ is a rational number.

I know how to show that specific numbers such as $2^{1/2}, 2^{1/3}, 3^{1/2}, etc.,$ are irrational, but what about the general form $m^{1/n}$?
7
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0answers
99 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
1answer
66 views

Is $12^{1/3}$ irrational?

Is $12^{1/3}$ irrational? Give a proof that justifies your answer So far I have: Suppose $12^{1/3}$ is rational.This means there exists integers a and b such that $12^{1/3} = \frac{a}{b}$ where ...
2
votes
1answer
54 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 ...
9
votes
2answers
142 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)- ...
0
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0answers
40 views

By induction, show that $\sqrt{p}\notin\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},\cdots,\sqrt{p_k})$ [duplicate]

By induction, show that $\sqrt{p}\notin\mathbb{Q}(\sqrt{p_1},\sqrt{p_2},\cdots,\sqrt{p_k})$ where $p_1,p_2,\cdots ,p_k,p$ are distinct primes. My try: For $k=1, ...
10
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1answer
86 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 ...
1
vote
5answers
51 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 ...
7
votes
3answers
146 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
votes
3answers
482 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 ...
-3
votes
2answers
77 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 ...
0
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4answers
76 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?
3
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2answers
62 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
45 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
votes
0answers
31 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
votes
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
56 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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vote
8answers
176 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 ...
5
votes
1answer
135 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 ...
7
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11answers
4k 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 ...
2
votes
2answers
66 views

Proof of irrationality without using contradiction

I'm just wondering if there exists proofs that certain numbers are irrational that do not begin by saying some like along the lines of "assume $k=a/b$ for integers $a$ and $b$" and then deduce a ...
1
vote
2answers
69 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: ...
1
vote
4answers
566 views

Suppose that x and y are irrational, but x + y is rational. Prove that x - y is irrational. [closed]

I can understand how it works in my head, I don't know how to prove it though.
0
votes
1answer
111 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
43 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 ...
9
votes
2answers
180 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 ...
1
vote
0answers
32 views

Proof Verification of Sum of Irrational Numbers

The chosen answer here claims that for rational $u$ and $v$, $u^{\frac{1}n} + v^{\frac{1}n}$ is rational iff $u^{\frac{1}n}$ and $v^{\frac{1}n}$ are both rational. However, the link to a proof seems ...
3
votes
2answers
150 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 ...
3
votes
1answer
68 views

An infinite sum based on the mod-parity of Euler's totient function

Let $\bmod( m,k )$ be the remainder when $m$ is divided by $k$: $0,1,\ldots,m{-}1$. Let $\phi(n)$ be Euler's totient function: the number of relatively prime numbers smaller than $n$. So for ...
7
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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3answers
83 views

Proof of $6 - \sqrt{2}$ is Irrational by Contradiction

What is a Proof by Contradiction, and how to prove by contradiction that $6 - \sqrt{2}$ is an irrational number?
0
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1answer
33 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 ...
1
vote
2answers
95 views

Solution of the equation $x^x=2$

Let $x$ be the solution of the equation $x^x=2$. Is $x$ irrational? How to prove this?
0
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0answers
55 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...
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votes
1answer
97 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 ...
0
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3answers
63 views

Prove $\sqrt[5]{672}$ is irrational

How would you prove $\sqrt[5]{672}$ is irrational? I was trying proof by contradiction starting by saying: Suppose $\sqrt[5]{672}$ is rational ...
-1
votes
3answers
88 views

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

How to simplify? $\frac{1}{1+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{7}} \frac{1}{\sqrt{7}+3}$
2
votes
0answers
40 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
48 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
184 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 ...
3
votes
0answers
57 views

When does the following construction generate a transcendental number?

Given $n\in[0,1]$ with base-b expansion $0.n_1n_2n_3\dots$, define $\Delta_b(n)$ to be the number with the following base-b expansion: $\huge{ 0.\underbrace{n_1}_{1^{st}\text{ ...
2
votes
0answers
71 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
40 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 ...
4
votes
5answers
1k 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 ...
2
votes
3answers
41 views

Prove the following based on number theory

$\log_5(2) \in \mathbb{R}\setminus \mathbb{Q}$ (irrational numbers). I know there is a question out there already for this but my problem is that I need to prove this using the fundamental ...
0
votes
1answer
40 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. ...