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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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How we calculate the value of $2^\pi$ without using calculator

How we calculate $2^\pi$? since $\pi$ is irrational how shall I calculate this? and can we write $$(2^\pi)(2^\pi)=(2^\pi)^2$$ and if yes what will be the condition, since $\pi$ is irrational no.
Mahendra Singh Adhikari
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Can I rationalize the denominator of $\frac{1}{\pi}$?

Can I rationalize the denominator of $\frac{1}{\pi}$?
user755960
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π≠a+b√2 when a and b are both rationals

I'm trying to prove that not all irrational numbers are of the form c=a+b√2 when a,b are rational. I'm thinking if I prove that π does not have this form, them there's an infinity of irrationals which can be written like c=a+bπ with a,b rationals…
Miguel Angel Andrade Velázquez
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Prove that $4^{1/3}+10^{1/2}$ is irrational

Show that $4^{1/3}+10^{1/2}$ is irrational. I start by assuming it to be rational and want to come to a contradiction $$ 4^{1/3}+10^{1/2} = r \\ \Rightarrow 4^{1/3} = r - 10^{1/2} \\ \Rightarrow 4 = (r-10^{1/2})^{3} \\ \Rightarrow 4 = r^{1/3} -…
Kirito
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Irrational Base Number System: Any Benefits?

I saw a YouTube video recently that mentioned the golden number, phi, being a viable base for a number system, given $\Phi^2 = \Phi + 1$. The result would be the units place representing multiples of $\Phi^0$, the next value left be multiples of…
Iter
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Is the "conjugate" of irrational numbers standard terminology?

In some high school text books I find theorems called "Irrational Conjugate Theorem" (or similar) related to irrational roots of rational polynomials. More precisely they call the number $p-\sqrt{q}$ the "conjugate" of $p+\sqrt{q}$ ($p$, $q$…
Ferenc Beleznay
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For real positive $a,b$ given $a^2b^2(a^2b^2+4)=2(a^6+b^6)$, show that at least one of the numbers is irrational

I think I am close to the answer, but still not sure how to finish it. $$a^2b^2(a^2b^2+4)=2(a^6+b^6)$$ I know I can rewrite that equation as $$(a^2-b\sqrt2)(a^2+b\sqrt2)(a\sqrt2-b^2)(a\sqrt2+b^2)=0$$ Thus $$a^2=\pm b\sqrt2…
aradarbel10
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Irrationals with repeated binary digits

Let r a real number. Let f(r) be a function of r such that in binary representation of r, the first digit is repeated once, the second digit is repeated twice, the third three times and this is continued ad infinitum. Couple of examples: r =…
We Pretty
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rational fractions and the negative sign

Say we have the expression $$\frac{a}{b}=\frac{a+3}{b-8}$$ When we cross-multiply the terms we end up with $$a(b-8)=b(a+3)$$ If we try $a=-3$ and $b=8$ in the previous expression we get $-3(8-8)=-8(-3+3)$ that results into $0=0$. As both sides are…
yCalleecharan
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Is it always possible match two non-equal closed form Algebraic Irrational Numbers (real or complex) so the product is a rational number?

(This is a question from an engineer and extremely naïve mathematician when it comes to the topic of irrational and transcendental numbers and the precise distinction between them.) I begin with two examples of what I mean by the title question: Let…
James Arathoon
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every positive rational number can be written as a finite sum of distinct numbers of the form 1/n where n is natural

every positive rational number can be written as a finite sum of distinct numbers of the form 1/n where n is natural I don't quite understand this question. Actually, what does distinct numbers of the form $1/n$ mean? Does it mean $m(1/n)$ or…
user8314628
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Prove that if $x^3 + 3x + 3$ is irrational then $x$ is irrational, by proving the contrapositive

I don't understand how to do this considering that the contrapositive of $x^3$ is irrational. For example $2$ to the cube root is irrational, but I am trying to prove that is is rational
chickinougat
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Periodic representation of pi in varying base system

Pi and e ar irrational numbers and cannot have a periodic representation in a fixed base number (binary, decimal, hex, etc). However, if you choose variable base like 1!, 2!, 3!,.. e becomes 1.1111..1.. there is a similar trick to select a “variable…
Stepan
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Irrational Numbers Expression

Say $x$ and $y$ are irrational numbers. I know that $x + y$ could be rational, for example $\pi$ and $-\pi$. Also, x/y could be rational, for example pi and 1/pi. But would $x + y + xy$ be irrational? I couldn't find an example where the…
IUissopretty
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What exactly is this question asking? (radix representation of rational numbers)

I'm not asking anyone to solve the problem, I just want to understand it. At first I thought I just had to change bases, but it seems that I have to do something else (and that's why they've given $M$ and $N$). The notations $N$, $M$ and $r$ are…
Mil3d
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