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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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 / ...
3
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2answers
90 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^+, ...
27
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12answers
1k 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?
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4answers
191 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 ...
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2answers
37 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 ...
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0answers
77 views

Irrational + Irrational = Rational? [closed]

Is it possible for two irrational numbers, when added, to be a rational number? The numbers should not be such that the irrational part can be cancelled. Without the criteria, I can obviously get ...
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4answers
43 views

Suppose $a$ is an irrational positive real number. Prove that if $x$ and $y$ are integers and $x\neq y$, then $(x + a)/(y + a)$ is irrational. [closed]

I need your assistance. Please provide a proof of the following proposition: Proposition: Suppose $a$ is an irrational positive real number. Prove that if $x$ and $y$ are integers and $x\neq y$, then ...
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0answers
64 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 ...
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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, ...
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9answers
7k 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 ...
4
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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 ...
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1answer
20 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 ...
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2answers
33 views

Can we prove $\sqrt6$ is irrational by the method of contradiction? [closed]

Can we prove that $\sqrt 6$ is irrational number using the meathod of contradiction?
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1answer
33 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)| ...
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4answers
627 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 ...
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1answer
43 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 ...
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2answers
33 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 $m^{1/n}$?
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0answers
95 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
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1answer
65 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
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1answer
45 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 ...
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2answers
136 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)- ...
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0answers
36 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, ...
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1answer
83 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
47 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
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4answers
131 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 ...
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3answers
475 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
74 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
63 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
60 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 ...
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2answers
44 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 ...
4
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4answers
141 views

How to prove that the value of $e$ is irrational without using the number $e$ itself [closed]

Recently I have tried to prove that the value of $e$ is irrational without using the number $e$ itself. I have seen that the number $e$ can be expressed as $$\lim_{n\to\infty}(1 + 1/n)^n;$$ however, ...
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0answers
29 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 ...
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1answer
74 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.
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0answers
52 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
168 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
124 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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11answers
3k 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
64 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 ...
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2answers
62 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
488 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.
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1answer
101 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
35 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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0answers
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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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0answers
29 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
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2answers
147 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
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1answer
66 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 ...
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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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3answers
81 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?
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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 ...
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2answers
94 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?