Mathematics Stack Exchange
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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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If $n$ is any positive integer, prove that $\sqrt{4n-2}$ is irrational

If $n$ is any positive integer, prove that $\sqrt{4n-2}$ is irrational. I've tried proving by contradiction but I'm stuck, here is my work so far: Suppose that $\sqrt{4n-2}$ is rational. Then we have $\sqrt{4n-2}$ = $\frac{p}{q}$, where $ p,q \in…
meiryo
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Proving that $\sqrt{2}^{\sqrt{3}}$ is irrational

I know how to prove that $\sqrt{2}$ and $\sqrt{3}$ are irrational. But, I am not sure in this case. $x=\sqrt{2}^{\sqrt{3}}$ gives me equation $\ln {x}-\frac{\sqrt{3}}{2} \ln{2}=0$ which is not polynomial, so I can not use theorem about rational…
Someone347
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irrational number: etymology of the word

One usual meaning given to the expression "irrational number" is "number not logical or reasonable". In particular in Spanish, where the usual term is "número irracional" (without logic, madness). This goes against what is the main characteristics…
pasaba por aqui
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Irrationality measure.

I would like someone to give me a definition of what irrationality measure is, I have stumbled over several definitions which may be equivalent but as I lack understanding I cant see this correlation. So I am looking for a definition and also a…
user117449
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probability that a number defined by a random process is irrational

What if we write $0$. and then throw a coin and depending on the result continue the number with 1 or $0$ and continue this process indefinitely. It is clear that the result of this procedure is a real number. There is an infinity of infinite…
Adam
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Can we have a mirror image of an irrational decimal?

Is it possible to have a number that extends to the left of the decimal point in mirror image of an irrational number? Such as <...95141.30000...>, to write pi as a mirror image.
user12404
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For what values of $n$ is $\ln(n)/\pi$ irrational?

For what integers $n>1$ is $\ln(n)/\pi$ irrational? Clearly, if $\ln(m)/\pi$ and $\ln(n)/\pi$ are rational then $\log_mn$ must be rational. In particular, if there is prime dividing $m$ but not $n$, then $\log_mn$ is irrational, so one of…
Jakobian
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Why decimal form of $\frac{A}{2^m\times5^n\times3^p\times7^q\cdots}$ has both repeating and non repeating numbers?

I read in my book that, decimal form of the fraction $\cfrac{A}{\underbrace{2^m\times5^n}_{\text{at least one}}\times\underbrace{3^p\times7^q}_{\text{at least one}}\times\cdots}\quad(A\in\mathbb{N})$, has both repeating numbers and non repeating…
Etemon
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Mystery about irrational numbers

I'm new here as you can see. There is a mystery about $\pi$ that I heard before and want to check if its true. They told me that if I convert the digits of $\pi$ in letters eventually I could read the Bible, any book written and even the history of…
diff_math
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Irrationality of some specific numbers

Is $(\sqrt 2)^\sqrt3+(\sqrt 5)^\sqrt{7}$ an irrational number? is there any result about rationality\irrationality of numbers of the form $a^b+c^d$ where $a,b,c$ and $d$ are well-known irrational numbers (like $\sqrt{2}, \pi$ and...)
Payam Seraji
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Prove or disprove: For all positive integers $ n$ , $\sqrt[3]{n}+\sqrt[3]{n+1}$ are irrational numbers.

Prove or disprove: For all positive integers $ n$ , $\sqrt[3]{n}+\sqrt[3]{n+1}$ are irrational numbers.
tianzhidaosunyouyu
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Deleting digits from an irrational number

Is it true that by deleting infinitely many appropriate digits out of the decimal representation of any positive irrational number, we can always get back the original number?
user327929
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how many numbers of irrationality measure $x$

Does there exist $x>2$ such that uncountably many reals have irrationality measure x? Must there exist at least one number of irrationality measure $x$? related question on sets of constant irrationality measure
Angela Pretorius
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Proving $\frac{\arccos\frac15}\pi\not\in\Bbb Q$

How would one prove $$\frac{\arccos\frac15}\pi\not\in\Bbb Q$$ Fiddling around with numbers hasn't led me anywhere: Suppose $\frac{\arccos\frac15}\pi\in\Bbb Q$, suppose it is equal to $\frac ab$. Then $\arccos\frac15=\frac{a\pi}b$ and thus…
konewka
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Are there any non-trivial counterexamples to the non-closure of the irrational numbers over addition?

It is trivial to show that the set of irrational numbers is not closed under addition. Just choose an irrational number $p$ and add it to its additive inverse $-p$ to get $0\in\mathbb{Q}$. However, I have yet to see a (non-trivial) example of a…
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