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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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for a rational number $a$ and an irrational number $b$ , if $ab$ is rational or $\frac{a}{b}$ is rational can we say that $a=0$?

I guess product or division of a rational and an irrational number can be rational and that also only $0$, when the rational number is $0$. In all other cases it will be irrational . Please correct me if I am wrong .
Sameer Nilkhan
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Why is $\sqrt 2 \sqrt 3 = \sqrt{2\cdot 3}$?

It seems very obvious that $\sqrt 2 \sqrt 3 = \sqrt 6$. But if we think that $\sqrt 2$ is irrational and that $\sqrt 3$ is also irrational, how can we prove that product of two irrational numbers is equal to another irrational number? Please…
Sameer Nilkhan
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Is $\sin(x)$ an irrational number?

I was wondering, is $\sin(x)$ an irrational number? I know that for $0$ it is $0$, so rational, but from basic view.
naruto25
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Predicting digit sequences in irrational numbers?

I'm trying to determine to what degree digits can be predicted in irrational numbers in general. I learned about normal numbers via this prior question: Predicting digits in $\pi$, which seems to imply that if numbers can be normal, digits may not…
DukeZhou
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$-(\pi^5)/(2\sqrt{2})\cdot(\cot(\pi/\sqrt{2}))$ is irrational? How to prove it?

Is $-\frac{\pi^5}{2\sqrt2}\times\cot(\frac{\pi}{\sqrt2})$ irrational? I have an idea, and this is the last question for this idea. Pi, and square root of 2 is irrational numbers, but these expression, i don't know. Please, help! Thanks for the…
Kristóf Lőrincz
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How to show an irrational equation have no solution?

There are different cases, one of them is when I have a negative $\sqrt{x}$ so I know there is no solution because $\sqrt{x}$ can't be negative. However there is another case where I just can't isolate $x$, $x$ is always depending on $\sqrt{x}$ (and…
ThomasRift
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Show that $\sqrt{abc}$ is irrational if $a, b, c$, and $\sqrt{a} + \sqrt{b} + \sqrt{c}$ are irrational.

Assume $a$,$b$,$c$ are the irrational numbers, and $\sqrt a + \sqrt b + \sqrt c$ is irrational number. Show that $\sqrt{abc}$ is irrational number. Please help me this problem, thank you for watching!
Chris Li
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Surds - Finding square roots.

To find square root of surd like this : $a+\sqrt{b}+\sqrt{c}+\sqrt{d} $ We put it equal to $\sqrt{x}+\sqrt{y}+\sqrt{z}$ To find the square root of : $21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15} $ can we put this equal to $ \sqrt{x}+\sqrt{y}+\sqrt{z}$ please…
Sachin Sharmaa
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Sequences of Irrational numbers

Let $ \alpha $ is fixed irrational number. Let $ [-h,h] $ be an interval. Is it true that for any sequence of irrational numbers $ \{h_n\} $ converges to zero in $ [-h,h] \, $ both $ \, \alpha + h_n \, \mbox{and} \, \alpha - h_n \, $ are always…
Uswadkar Prashant Vasantrao
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How to write the contrapositive of this statement..?

"There is $x \in \mathbb{Q}$ such that $x+y \notin \mathbb{Q}$ implies $y \notin \mathbb{Q}$." Would it be "$y \in \mathbb{Q}$ implies there is $x \in \mathbb{Q}$ such that $x+y \in \mathbb{Q}$"? ..or an $x \notin \mathbb{Q}$?
user607212
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Do the digits(after comma) of the irrational number $\sqrt{2}$ contain all natural numbers without jumping order?

If we mention about $\sqrt{2}=1.41421356$ It contains 1,2,3,4,5,6,41,414,4142,41421 But it does not contain 4121 $\sqrt(2)$ does not contain repetitive patterns and it is not transcendental number(0,10010001...)(which contains only specific digits)…
etfwrtfrew etghtrth
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Is there a non trivial set of pairs of irrationals that give a rational when multiplied

I'm wondering if there exists a non trivial set of pairs of irrationals that gives a rational when multiplied together. What I mean with trivial is for instance the set of pairs composed of an irrational and one of its multiplicative inverse :…
user3344867
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Simplification of Irrational Numbers

Just a quick question i'm currently confused, would be grateful if you anyone could provide full working out. "Without resorting to the use of a calculator or computer, find a simpler representation for each of the numbers below:" $\sqrt{2+\sqrt3} -…
RalphCarroll
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How do you think of irrational numbers?

What are some ways by which you can characterise an irrational number? The basic way is as those real numbers inexpressible as integral fractions; another is as those reals with non-periodic decimal expansions; another would be as quantities…
Allawonder
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Neighborhood of an Irrational number.

Say, $c$ with $c>0$, is an irrational number, what I need to prove is, any $\delta$-ngbhd of $c$ i.e. $V_{\delta}(c)$ contains a finite number of rational numbers. I am not sure if the statement is correct or not. Could anyone comment/provide a hint…
User9523
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