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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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Can an irrational number raised to an irrational power be rational?

Can an irrational number raised to an irrational power be rational? If it can be rational, how can one prove it?
John Hoffman
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Is there a "positive" definition for irrational numbers?

An irrational number is a real number that cannot be expressed as a ratio of integers. Is it possible to formulate a "positive" definition for irrational numbers? A few examples: An irrational number is a real number that can be expressed as... An…
barak manos
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Rational + irrational = always irrational?

I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 + irrational number is irrational, then it stood to…
gator
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Is it possible to represent every irrational number as a (limit of) an infinite sum of rational numbers?

For instance, we can certainly represent π in this fashion. $$ \frac{\pi}{4} \;=\; \sum_{n=0}^\infty \, \frac{(-1)^n}{2n+1} .\! $$ $\ln(2)$ is also irrational. And even that can be represented as an infinite sum of a sequence of rational…
shreedhar
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Why are irrational numbers such a big deal?

Why is it such a big deal that some numbers are irrational? It means they can't be represented as integer fractions. Cool. But almost all numbers satisfy that property. So why is it that, for example on $\pi$'s wikipedia page, already in the third…
Jaood
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Expansion of $(1+\sqrt{2})^n$

I was asked to show that $\forall n\in \mathbb N$ there exist a $p\in \mathbb N^\ast$ such that $$(1+\sqrt{2})^n = \sqrt{p} + \sqrt{p-1}$$ I used induction but it wasn't fruitful, so I tried to use the binomial expansion of $(1+\sqrt{2})^n$ but it…
Meadara
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Prove or disprove that $2^{\frac{\pi}{2}}$ is an irrational number.

Problem Prove or disprove that $2^{\frac{\pi}{2}}$ is an irrational number. My Try According to our mathematical intuition, we may want to apply Gelfond–Schneider theorem, which states that if $\alpha,\beta$ are two algebraic numbers, where…
mengdie1982
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$\sqrt{2}\notin\mathbb{Q}$ but ...

Ok, it's easy to prove that prime roots are irrational (i.e. $ \sqrt{p} \not\in \mathbb{Q}, \text{ if } p \in \mathbb{P} $) Consider $ \sqrt{2} $. We can quickly prove that $ \sqrt{2} \not\in \mathbb{Q} $. Proof: (by contradiction) Assume that…
Stephen
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Why are $e$ and $\pi$ so common as results of seemingly unrelated fields?

I'm sure this gets asked all the time but I swear I googled with no useful result. What I'm looking for is a reasonably intuitive answer. Those two constants have some pretty interesting properties. $\pi$ is often used in geometry while $e$ is for…
Luka Horvat
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Is there a proof that $\pi$ is an irrational number?

Most math texts claim that $\pi$ is an irrational number. However, I'm having a little bit of trouble understanding that. Since nobody has calculated all of the digits of $\pi$, how can we know that either: one of the digits repeats (as in…
Nathan Osman
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If the square root of two is irrational, why can it be created by dividing two numbers?

I'm doing pre-calculus and got a bit caught on this question… I looked online and it said that a number is rational if it can be the quotient of two integers. So I did this: I know that it is irrational because if this was actually rational,…
Rob Dawson
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Irrational Numbers and ways to represent all of them

I just started studying "Analysis" and it appears one of the first topics all textbooks goes through, is making the reader understand that there are "gaps" in Q. Therefore, we need numbers that are irrational to fill in the gaps. And those numbers…
Snowball
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Irrationality of $\sum\limits_{n=1}^{\infty} r^{-n^{2}}$ for every integer $r > 1$

In the preface to Introduction to Algebraic Independence Theory Yuri V. Nesterenko mentions the series $$f(r) = \sum_{n=1}^{\infty} \frac {1}{r^{n^{2}}}$$ which was introduced as an example by Joseph Liouville in 1851, who proved that $f(r) $ is…
Paramanand Singh
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Deciding whether a number is rational (2 examples)

1) Prove that number irrational $\sqrt{7-\sqrt{2}}$ I created a polynomial $x=\sqrt{7-\sqrt{2}}$ so $P(x)=x^4-14x^2+47$ and since $47$ is prime we check $P(x)$ for $ {1,-1,47,-47}$ and since all of them are $P(x)\neq0$ it means our number is…
Gregor
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Irrationality proof by fast converging series?

I read here https://www.mathpages.com/home/kmath455/kmath455.htm that $\sum_{n=1}^\infty \frac{1}{d_n}$ is irrational if $d_{n+1} > d_{n}^2$ for all $n > N_0$. Can we prove $\pi$, $e$ or some other numbers irrational by creating a series for them…
user58512
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