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Questions tagged [irrational-numbers]

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.

An irrational number is a real number that cannot be expressed as a quotient of two integers, i.e. cannot be expressed in the form $\dfrac{a}{b}$, with $a,b\in\mathbb{Z}$. We write $\mathbb{I}=\mathbb{R}\setminus\mathbb{Q}$.

Some examples of irrational numbers are $\sqrt{2}, e, \pi$ and $\zeta(3)$.

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On the cardinality of rationals vs irrationals

I understand that there are infinitely many more irrational numbers than there are rationals - there are many ways in which one can intuitively understand this. However, consider the following: LEMMA: For every pair of distinct irrational numbers,…
Chris
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Prove that $5^{1/3}-3^{1/4}$ is irrational

Prove that $5^{1/3}-3^{1/4}$ is irrational. Let $x=5^{1/3}-3^{1/4}$. I started by: \begin{align} (x+3^{1/4})^3&=5\\ x^3+3^{3/4}+3x\cdot3^{1/4}5^{1/3}&=5\\ 3^{3/4}+3x\cdot3^{1/4}5^{1/3}&=5-x^3 \tag{1} \end{align} Now I wished to prove by…
Gaurang Tandon
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Is $\pi$ a property of mathematical expressions or algorithms?

I was trying to think about what $\pi$ actually is. There are a lot of ways to get $\pi$ for example $4(1-\frac{1}{3}+\frac{1}{5}-\cdots)$. But there is no one way to define it. On the other hand a fraction like $\frac{1}{2}$ also has multiple…
zooby
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If $a+b$ is an irrational number, is $a-b$ an irrational number, too?

Question 1: If $a+b$ is an irrational number. Is $a-b$ an irrational number, too? Question 2: If $\cos(a)-\sin(a)$ is irrational, Is $\sin(a)-\cos(a)$ irrational, too?
jiun
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Irrational equation with 2 terms

Let $ n \in \mathbb{N}, n \ge 2$. Prove that $\sqrt[n] {\sqrt{2018}+\sqrt{2017}}$ $+$ $\sqrt[n]{\sqrt{2018}-\sqrt{2017}} $ $\in \mathbb{R}/\mathbb{Q}$ Any ideas? I noticed that first member multiply with the second is equal with $1$. I tried to…
Radu Andrei
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Prove that $\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}$ is rational if and only if $a^2-b$ and $\frac{1}{2}(a+\sqrt{a^2-b})$ are square.

Let $a,b \in \mathbb{R}$ such that $a^2\geq b$. Prove that $\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}$ is rational if and only if $a^2-b$ and $\frac{1}{2}(a+\sqrt{a^2-b})$ are square. $\Rightarrow$) $\begin{align*} \sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}&=…
Emma Wool
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Prove an infinite sum is irrational

I'm trying to prove that $$ \sum_{k=1}^{\infty} 7^{-k!} $$ is irrational but I'm so lost. Any tips for where to begin, thanks in advance.
Martin Carrasco Filippi
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Proof of Irrationality of a non Recurring Decimal

How to prove that 0.101001000100001..... is irrational? There's the fact that it is non recurring but is there any mathematical proof like that we give for square root of two?
Petra
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(How to/Can I) show irrational numbers?

This might sounds stupid, but I really don't know can I show Irrational numbers in proves? And if so, how to show it? For example, when I want to show Rational numbers, I do this: $\frac{m}{n} $ , $m, n $ are integer $,$ $n\ne0$ Can I do something…
Mostafa Farzán
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What is the term for 'PI-indexing'?

As a teenager - the concept of irrational numbers fascinated me. The idea that all possible numbers existed in PI. From that I reasoned that any piece of data you have now also existed in PI somewhere. For a moment I thought that this could lead to…
hawkeye
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Proof of an irrationality criterion

I have attached a proposition whose proof I don't understand at two points. Here are my questions: Why do we have $|a_{0n}+\theta_{1}a_{1n}+\dots+\theta_{k}a_{kn}|<(\rho-\varepsilon)^{-n}$ for sufficiently large $n$? Why do we have…
user337073
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Rationalize a surd $\frac{1}{1+\sqrt{2}-\sqrt{3}}$

How can I rationalize the following surd $$\frac{1}{1+\sqrt{2}-\sqrt{3}}$$ What would be the conjugate of the denominator
mahes
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Square root of the product of consecutive natural numbers is irrational

Prove that for all $n\in\mathbb{N}$ the number $\sqrt{n(n+1)}$ is irrational. My first move would be: Let's assume that it's not, that it $\sqrt{n(n+1)} = \frac{a}{b}$, where $a,b\in\mathbb{N}$ and $a,b$ are coprime. Then: $$ n(n+1)=\frac{a^2}{b^2}…
Mat Dyl
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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 $\gcd(a,b) = 1$. Then, $12 = \frac{a^3}{b^3}$ so $a^3…
user236122
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Dedekind's method for irrational number

I am now reading the definition of irrational number, which we can describe by the following terms: suppose that we have divided all rational numbers into two classes, a lower class and an upper class, such that every number of the lower class is…
dato datuashvili
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