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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Prove the following statement: $\forall r\in \mathbb{R^+}$,if r is irrational then $\sqrt r$ is irrational

Prove the following statement: $\forall r\in \mathbb{R^+}$,if r is irrational then $\sqrt r$ is irrational My attempt: if suppose $\sqrt r $ is rational then there exists $p,q \in \mathbb{R^+}$ such that $\frac{p}{q}=\sqrt r $ where p and q…
user271336
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Show that the number $\sqrt{2^1}+\sqrt{2^2}+\cdots+\sqrt{2^n}$ is not rational for any integer $n>0$

I've got the first part of the proof. So we know that the odd exponents will give us an irrational answer, and the even ones will give us a rational answer. Therefore we have an alternating sum of rationals and irrationals. I can prove that an…
Gerard L.
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Prove that $x+y$ is irrational if x and y are irrational and positive.

I haven't seen really a straightforward proof towards this question. All of them regarding this topic focus on the fact that $x+y$ can be rational even if x and y are irrational because you could set y as the negative of x, but there isn't really…
Gerard L.
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Irrationality of the twelfth root of 2

How can i prove that $\sqrt[12]{2}$ is irrational number? I'm trying: $$\sqrt[12]{2} = \frac{p}{q}$$ where $p$, $q$ are integers it follows that : $$p^{12} = 2q^{12} $$ What is argument of irrationality in this case? From what we know that the…
Krzysztof Michalski
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Online tool to check if number is rational or irrational?

I am new to this forum. I've been programing for some time, and now starting my engineering degree. I am trying to find an online utility that will help me grasp the concept of irrational numbers (summary, multiplication, devision etc.) through…
vondip
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Arbitrary Sequence of Digits in Irrational Number

What are numbers in which we can find arbitrary sequence of digits (in a certain base-$n$ expansion)? I know that $0.123456789101112131415\cdots$ does (and its analogues in other bases), but does this property hold for some more familiar numbers…
finnlim
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irrationality of $\sum_{k=1}^{n} k^{\frac{1}{m}}$

For arbitrary $n \geq 2$ and $m \geq 2$, is $$\sum_{k=1}^{n} k^{\frac{1}{m}}$$ an irrational number?
this_is_a_banana
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Are irrational numbers irrational by nature?

I remember hearing an interesting theory once, I don't know the source. Since there are some numbers that are precisely expressible in decimal notation that repeat in a binary base, and vice versa, perhaps there exists a base in which irrational…
Chuck
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Some questions about proofs of irrational numbers

I have some questions about some things I want to clarify in regard to basic questions that ask to show that roots are irrational, for example $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{6}$. To me, I think there are a few little lemmas/thereoms I am using…
Quality
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Can length of line segment have non-terminating decimal form value?

Premise 1: All straight line segments have the value of length equal to the numerical value of the end point, provided the starting point of the line is assigned the numerical value zero. Premise 2: Lengths of certain value exist if they can be…
Sensebe
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How do I check if intervals can be nested?

Hi I'm in grade 10 and I'm doing nested intervals. Under my current circumstances I am unable to have a teacher, so I'm self teaching with the help of textbooks. I understand very, very basic intervals, but I've come to a question where it wants to…
Maddy. P
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Help With a proof (Irrational Number)

Prove the following statement by proving its contrapositive: if $r$ is irrational, then $r^\frac{1}{5}$ is irrational. Its contrapositive will be: If $r^\frac{1}{5}$ is not irrational, then $r$ is not irrational. How can I prove the contrapositive…
Untimely Answers
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Square root of an odd composite being irrational

Is there an odd composite number $n$ such that $\sqrt{n}$ is irrational?
user132181
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Looking for irrational Numbers Proof

$a,b,c,~$and $d$ are rational numbers. $b>0$ and $d>0$ the $\sqrt{b}$ and the $\sqrt{d}$ are both irrational. if $a+\sqrt{b}=c+\sqrt{d}$ show that $a = c$ and $b = d$. I know that a=c and b=d intuitively, but I'm not sure how to prove it.
Nikita
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