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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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Proving that $\sqrt{3} +\sqrt{7}$ is rational/irrational

I took $\sqrt{3}+\sqrt{7}$ and squared it. This resulted in a new value of $10+2\sqrt{21}$. Now, we can say that $10$ is rational because we can divide it with $1$ and as for $2\sqrt{21}$, we divide by $2$ and get $\sqrt{21}$. How do I prove…
gbox
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a square root of an irrational number

I wonder if a square root of an irrational number is always irrational? I would tend to think that yes, but I can´t think of any justification. Also there are cases which are rather hard to decide like sqrt(Pi).
Adam
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Irrationality Preserving Operations

We all know about rationality preserving operations. That is, if $r,s\in\mathbb{Q},$ we have that $r+s,r-s,rs,\frac{r}{s(≠0)}\in\mathbb{Q}.$ I was wondering if there are irrationality preserving operations. I know that addition isn't such an…
aqualubix
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How to get $\sqrt{2}$ is irrational with a lower irrational band

I found this solution about the constructive solution of irrationality of sqrt 2 on Wikipedia. In the last paragraph it says: This gives a lower bound of $\frac{1}{3b^2}$ for the difference $\left|\sqrt{2} − \frac{a}{b} \right|$. I didn't…
Jacob Martina
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approximating $1/{\sqrt{2}}$ by rationals

If $0<\ \frac{p}{q}<1$ is a rational number, then prove that $$\left|{\frac{1}{\sqrt{2}}}-{\frac{p}{q}}\right|> {\frac{1}{4q^2}}.$$ I found a proof using Liouville's inequality working with $f(x) = 2x^2 - 1$, but am wondering if there is a direct…
student
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Can $y = \alpha x$ be solved for $\alpha$ irrational if the integer parts of $x$ and $y$ are unknowns?

Take three real numbers $x$, $y$, and $\alpha$, of which $\alpha$ is a given irrational number. Can $$y = \alpha x$$ be solved if the fractional parts of $x$ and $y$ are known (given) but their integer parts are unknown? Note: this is a restatement…
Arc
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is there any way to represent irrational numbers with a finite amount of integers?

I know that rational numbers can be represented with two integers $\frac{a}{b}$. But is there any way to represent irrational numbers with an finite amount of integers? My best guess is $\frac{a}{b} ^ \frac{c}{d}$. it can represent any root of any…
gdor11
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Direct proof for the irrationality of $\sqrt 2$.

Prove that $\sqrt 2$ is irrational using direct proof. I have seen TONS indirect proofs (e.g. proof by contradiction) for it, and people say that it's difficult to proof this directly. So is this impossible? Thank you.
JSCB
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Proof that an equation is irrational

I hope you're keeping safe and well. I stumbled across this problem and wondered whether you could help. Show that $\left(a+\sqrt b\right)\left(a-\sqrt b\right)^3$ is irrational if $a$ and $b$ are NOT square numbers. Thank you so much for your…
Pac-Man
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Do the decimal digits of $\pi$ and $\sqrt{2}$ coincide infinitely often?

I suspect the answer to the title question is "yes," but can it be proved? One would expect digits match in one tenth of cases.
Bernardo Recamán Santos
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how or why the method long division for finding square root of a number works

I was wondering how this method developed and also how it works to give the square root of a number upto a large number of decimal digits . Please guide me.
Sameer Nilkhan
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what was developed first , the mthod or the result , of finding square root of a natural number?

I was wondering , whether the methods for finding square root of a number were developed first and then by using them value of square root of any number was found OR first the value of square root of any number was found manually by hit and trial…
Sameer Nilkhan
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Rational (non-negative) linear combinations of square roots of non-square naturals cannot change the status of irrationality, can they?

I arrived at an idea of considering rational linear combinations of square roots of non-square naturals and of the irrationality of those combinations. So suppose that we have some $n$-tuple $(\sqrt{a_1},...,\sqrt{a_n})$ where $a_1,...,a_n$ are…
user716491
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Is the sum $\sum_{n=1}^{\infty}\frac{1}{1+n^2}$ irrational?

$$\sum_{n=1}^{\infty}\frac{1}{1+n^2}\approx1.07169$$ I have viewed the proof of the irrationality of $e$ and $\frac{\pi^2}{6}$ but no idea with this one.
yuanming luo
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Is a rational to the power of itself irrational?

Take two coprime natural numbers $a>0$ and $b>1$ Let $x = \frac ab$. Is it guaranteed, that $y = x^x$ is irrational? If not, which properties do $a$ and $b$ or $x$ need? Edit 1: Thank you @mathworker21 and @fleablood. Now its easy to prove that…
Kinheadpump
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