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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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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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Can two integer polynomials touch in an irrational point?

We define an integer polynomial as polynomial that has only integer coefficients. Here I am only interested in polynomials in two variables. Example: $P = 5x^4 + 7 x^3y^4 + 4y$ Note that each polynomial P defines a curve by considering the set of…
Till
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Is there a proof that $\pi \times e$ is irrational?

A little reading suggests: It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is transcendental. If we just want irrationality rather than…
idmercer
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How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?

I was wondering... $1$, $\phi$ and $\frac{1}{\phi}$, they have something in common: they share the same decimal part with their inverse. And here it comes the question: Are these numbers unique? How many other members are in the set if they exist?…
Carlos Toscano-Ochoa
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There's something strange about $\sum \frac 1 {\sin(n)}$.

Clearly, $$\sum_{n=1}^\infty \frac 1{\sin(n)}$$ Does not converge (rational approximations for $\pi$ and whatnot.) For fun, I plotted $$P(x)=\sum_{n=1}^x \frac 1{\sin(n)}$$ For $x$ on various intervals. At first, I saw what you might…
Descartes Before the Horse
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Elementary proof that the limit of $\sum_{i=1}^{\infty} \frac{1}{\operatorname{lcm}(1,2,...,i)}$ is irrational

Show that the infinite sum $S$ defined by -$$S=\sum_{i=1}^\infty \frac{1}{\operatorname{lcm}(1,2,...,i)}$$ is an irrational number. I found this question while reading 'Mathematical Gems' by Ross Honsberger. After pondering over it for nearly an…
TheBigOne
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How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? How does it compare to other irrational numbers…
Sean
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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?
user242891
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How are first digits of $\pi$ found?

Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am talking about accurate digits by either…
Confuse
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Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient criteria for irrationality. When applying some of…
Chazz
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Linear independence of the numbers $\{1,e,e^2,e^3\}$

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$? The proof should not use that $e$ is transcendental. $e:$ Euler's number. $\{1,e,e^2\}$ is linearly independent over $\mathbb{Q}$ Any hints would be…
felipeuni
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Is Area of a circle always irrational

I have learned that $\pi$ is an irrational quantity and a product of an irrational number with a rational number is always irrational. Does This imply that area of a circle with radius $r$, which is $\pi$.r$^2$ is always an irrational quantity?…
deepa kapoor
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ArcTan(2) a rational multiple of $\pi$?

Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$. Of course the sum of these angles is $90^\circ = \pi/2$. I would like to know if…
Joseph O'Rourke
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Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question. Now, his inital solution was like this: let's take a…
Armen Tsirunyan
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Is this formula for $\frac{e^2-3}{e^2+1}$ known? How to prove it?

I found an interesting infinite sequence recently in the form of a 'two storey continued fraction' with natural number…
Yuriy S
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