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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 $\zeta(3)$.

2478 questions
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Irrationality of $x$ if $x > 1$ and $x^x = 2$

Show that if $x>1$ and $x^x=2$, then $x$ must be irrational. I know you have to show that it cannot be reduced into a form $\frac pq$, but get stuck with quite ugly algebra.
Mals T
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How to prove that $x=0.1234567891011\dots $ is irrational?

I'm in $9th$ class and I was wondering how to solve this problem. I only know how to prove that $0.1010010001\dots$ is irrattional.
Andrew
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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?
Erivelton Geraldo Nepomuceno
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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.
DaGuruDot
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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. coordinates $(p,0), (0,q)$. Within this lattice, we…
Anthony
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Is there a rational way to conceptualize an irrational number?

This is a request for help, not an attempt to challenge anything. Since $\pi$ is irrational, this tells me that it's impossible to express the distance around a circle in terms of the distance accross. That boggles my mind, but maybe it should…
Aaron Anodide
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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 Golden Ratio if: $${{a + b} \over a} = {a \over b} =…
Michael Lee
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A positive integer with is not a perfect square is a product of distinct prime factors

This was used as part of the explanation for the following question, but I don't see why it is true. How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect square?
trebob
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Problem in the proof of root 2 is irrational

The standard way we prove that the square root of $2$ is irrational is the following: Let us assume the square root of $2$ is rational and is equal to $\frac{a}{b}$ where $a$ and $b$ are co-prime. ∴$\sqrt{2} = \frac{a}{b} \Rightarrow…
RON
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How can Irrational numbers go on forever>

I understand what irrational numbers are and how they were first proved by the ancient Greeks. My question arsis when thinking of how a length of a line can be the square root of two if by definition the magnitude of root 2 is unending. Yet there is…
CatsOnAir
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does the long division method gives the exact value of square root of any non perfect square natural number?

does long division method or the other manual algorithms that are used to calculate square root of any number , gives the exact value of square root of any number OR they just approximate it as accurately as they can ? Please guide me.
Sameer Nilkhan
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Making Rational Expressions

Find all positive integers a and b such that $$ \frac{\sqrt2 + \sqrt a}{\sqrt3 + \sqrt b}$$ is rational. I tried Equating to some number r and squaring it, and the answer is a= 3 b =2, but im not sure if those are all
SuperMage1
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Help proving the following proposition

Prove that for each real number x, $$(x+\sqrt2)$$ is irrational or $$(-x+\sqrt2)$$ is irrational. Now I honestly just don't know where to even go with this one. I tried to do a proof by contradiction but I couldn't get anywhere and I've also…
Nathaniel
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Let $\alpha = 0.1011011101111\ldots$ be a given real number written in base 10

Let $\alpha = 0.1011011101111\ldots$ be a given real number written in base $10$, that is, the n-th digit of $\alpha$ is $1$, unless n is of the form $\frac{k(k+1)}{2}-1$ in which case it is $0$. Choose all the correct statements from…
Abdul Jaleel
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I am trying to prove $1-\sqrt{2}$ is an irrational number.

I tried to prove that $1-\sqrt{2}$ is rational: $1-\sqrt{2} = p/q$ $(1-\sqrt2)^2 = (p/q)^2$ $1^2 - 2\sqrt2 +2 = p^2/q^2$ $3-2\sqrt2 = p^2/q^2$ $3-p^2/q^2 = 2\sqrt2$ I know that $\sqrt2$ is irrational, but how should I proceed from here?
L.vieira
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