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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 to prove that a number is irrational

We write all postive whole integers after the comma, how do we prove that this is an irrational number? ($0.1234567891011121314...$)
user176791
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How to prove that $\cos(n)$ is irrational?

We know that $\cos(1)$ is real and transcendental (1). Then by using the fact that for every $n \in \mathbb{N}$ there exists a polynomial $P_n$ of degree $n$ with integer coefficients such that $\cos(nt) =P_n(cos(t))$ for all $t \in \mathbb{R}$ ,…
The very fluffy Panda
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Irrationality of e and farey fractions

How do we go about proving that $$[k! e] = k! \sum_{j=0 -> k} \frac{1}{j!}$$ I know that we could write $$e = \sum_{j=0 -> \infty} \frac{1}{j!}$$ But I don't see how that's going to help in the proof...
phoenix
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The irrationality of the square root of 2

Is there a proof to the irrationality of the square root of 2 besides using the argument that a rational number is expressed to be p/q?
hersch
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How can I prove that $\pi+\pi^2$ is irrational?

I am trying to prove that $\pi+\pi^2$ is irrational assuming that $\pi$ is transcendental. My work: I noticed that at least one of $\pi+\pi^2$ and $\pi-\pi^2$ is transcendental. (Because algebraic numbers are closed under addition.) However, similar…
Adam
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Suppose $n \in \Bbb N$ has a root that isn't whole, prove $\sqrt n$ is irrational

Suppose $n \in \Bbb N$ has a square root that isn't whole, prove $\sqrt n$ is irrational. The exercise wants me to prove it in the following steps: Suppose $\sqrt n$ is rational, then there exists a minimal natural number $p$ such that $p \sqrt n$…
Kantig Shoter
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Sufficient or necessary conditions for irrational numbers.

Today in class we saw that the sum of an irrational with a rational gives an irrational, but not necessarily that the sum of irrationals gives an irrational. And I was wondering what sufficient or necessary conditions you would have to satisfy two…
user1098937
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if $x^x$ is rational, is $x$ also rational?

Assume that $x$ is a real positive number. I tried to manipulate the equation $x^x=p/q$ with $p,q$ integers but couldn't really prove or disprove it. Is there any counter examples or this is true? Sorry if this is a trivial question.
Pinteco
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Can we calculate LCM of irrational numbers. Specifically for $\pi$

Irrational numbers can not have LCM. But in one of the books I read and LCM for thr triplet $( \pi/2 , \pi , 3\pi/2)$ was calculated and the answer was $3\pi$. If we can't find LCM for irrational numbers and pi is one,who is the result possible for…
Test Test
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How do you show this isomorphism?

How do you show that $\mathbb{R} \backslash \mathbb{Q} \cong \mathbb{N}^{\mathbb{N}}$? What is a good starting point in showing this?
realcoolnav
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decimal expansion which consists of the concatenation of the powers of $2$ is irrational

The real number in $(0,1)$ which has as its decimal expansion $$0.248163264128256512\cdots$$ must be irrational. I am trying to prove the assertion by contradiction. I assumed that this decimal expansion is eventually periodic. Thus after a finite…
student
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Finding certain digits of constructible irrational numbers.

Let $n\neq m^q$ and $round(\sqrt[q]{n})=m$ for some $\{m,n,q\}\in\mathbb{N}$. Then, is there any way to find the $n^r$th digit of $\sqrt[q]{n}$, for $r\in\mathbb{N}$? I once saw something about someone claiming to know the $2^{2020}$th digit of…
Zuter_242
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Existence of $p$ and $q$ such that $\big|\alpha -\frac{p}{q}\big| \leqslant \frac{1}{q^2}$ if $\alpha \in \mathbb{R}\backslash \mathbb{Q}$

Prove that if $\alpha \in \mathbb{R}\backslash \mathbb{Q}$, there exists one (actually an infinity) of integers $p$ et $q$ such that $\big|\alpha -\frac{p}{q}\big| \leqslant \frac{1}{q^2}$ and $0
math
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If the square of an irrational number r is irrational, can it be equal to a + br, where both a and b are rational

I am trying to evaluate such a statement: $$\forall r \in \mathbb{R} \setminus \mathbb{Q}: \ ({r^{2}} \notin \mathbb{Q} \implies \forall a \in \mathbb{Q} \ \forall b \in \mathbb{Q}: {r^{2}} \neq a + br)$$ but it seems to exceed my skills. Could you…
Arkadiusz Filipczyk
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Assumptions of the proof of irrationality

The proof of the irrationality of $$\sqrt{2}$$ begins by Let us suppose that $\frac{a}{b}$ is in its lowest terms, which is to say $a$ and $b$ have no common factor. Why can we suppose this?
Elena Greg
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