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.

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Extention of Euclid's GCD Algorithm. (The Art of Computer Programming, Volume 1, Edition 3, Section 1.2.1, Exercise 12)

Euclid's GCD algorithm which is used to find GCD of two input numbers, say, $c$ and $d$, needs the inputs to be positive integers. Exercise 12 provides an extension to this algorithm and allows $c$ ...
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0answers
34 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
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6answers
955 views

Alternate proof for “$\log_{10}{2}$ is irrational”

I need to prove that $\log_{10}{2}$ is irrational. I understand the way this proof was done using contradiction to show that the even LHS does not equal the odd RHS, but I did it a different way and ...
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1answer
46 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
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4answers
115 views

Prove that $\sqrt{n^2 + 2}$ is irrational

Question: Suppose $n$ is a natural number. Prove that $\sqrt{n^2 + 2}$ is irrational. From looking at the expression, it seems quite obvious to me that $\sqrt{n^2 + 2}$ will be irrational, since ...
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1answer
39 views

Sets of irrationals whose square contains a rational

Let $S$ be a subset of the irrationals. Also, lets assume that $S$ has infinitely many elements. My very general question is, under what non-trivial conditions does there exist an element $x\in S$ ...
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0answers
99 views

The cube of at least one irrational number is rational

I am supposed to prove the statement above. Here is what I have so far Suppose that the cube of at least one irrational number $n$, is rational. By definition of rational, there exists ...
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3answers
83 views

Question about $\displaystyle\sum_{n=1}^{\infty}\dfrac{|\sin(n)|}{n}$. [duplicate]

In several places on this site the sum $\displaystyle\sum_{n=1}^{\infty}\dfrac{\sin(n)}{n}$ has been discussed as a generalized alternating series, which therefore converges. I am curious about the ...
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1answer
64 views

Real number system

Is the set of rationals a subset of the irrationals? I always assumed it was, but given that irrationals are defined to be numbers that have an infinite, non-repeating decimal expansion, there cannot ...
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1answer
83 views

About the continuity of $f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k}$

Let $q: \mathbb{N} \to \mathbb{Q}$ be a bijection and denote the image of $k \in \mathbb{N}$ by $q_k$. Let $f: \mathbb{R} \to (0,1)$, $$ f(x) = \underset{q_k \leq x}{\sum_{k \in \mathbb{N}}} 2^{-k} ...
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votes
1answer
120 views

What's the value of tau?

I've seen $\tau$ on a title of a YouTube video and I need help knowing what the value is. I'm serious. I've never heard of the value. So, what is it? Also, is it rational or irrational (this part ...
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2answers
55 views

Can $x^{2q}$ be irrational for rational $x$ and $q$?

I think the answer to the question in the title is "yes", because $9^{2/3}$ is irrational by an argument similar to the accepted answer in this question. Or am I mistaken?
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3answers
103 views

Irrational numbers in between $n$ and $n+1$

Is the amount of irrationals numbers in between consecutive integers always the same? is this amount infinite?
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2answers
1k views

Proof that $\sin 10^\circ$ is irrational

Today I was thinking about proving this statement, but I really could not come up with an idea at all. I want to prove that $\sin 10^\circ$ is irrational. Any ideas?
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1answer
54 views

A positive integer with is not a perfect square is a product of distinct prime factors

This was used as part of the explanation for the following question, but I don't see why it is true. How to understand Apostol's proof of the irrationality of $\sqrt{n}$ if $n$ is not a perfect ...
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0answers
55 views

There are infinitely many $\frac{m}{n}\in\mathbb Q$ such that $|x- \frac{m}{n} |<\frac{1}{n^2}$

Let $x$ be an irrational number. Prove that there are infinitely many rational numbers $\frac{m}{n}$ such that $$|x- \frac{m}{n} |<\frac{1}{n^2}$$ It's clear that $-1<1/n^2 <1$ with ...
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1answer
193 views

Understanding the proof of “$\sqrt{2}$ is irrational” by contradiction.

I have some difficulties in understanding the proof of "$\sqrt{2}$is irrational" by contradiction. I am reading it in 10th class(in India) Mathematics book( available online, here ) This is the ...
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0answers
83 views

What is the name of this proof of, “$\sqrt{2}$ is irrational”?

Usually the proof of $\sqrt2$ is irrational is done by contradiction(e.g. here), but I found another similar but short proof in the book "Beginning Algebra for College Students" by Lloyd Lincoln ...
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votes
1answer
68 views

A field between $\mathbb{Q}$ and $\mathbb{R}$ ?

I really have trouble understanding a task. We've got $p\in$ P, while P are all prime numbers. Now we construct a field $$\mathbb{Q}[\sqrt{p}]:=\{x+y\sqrt{p}:x,y \in \mathbb{Q}\}$$ The Task is to ...
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1answer
55 views

Can any root, such as a square root or a cube root, be rational?

I've heard of this and most roots are irrational such as $\sqrt{8}$ and $\sqrt[3]{25}$. So, can any of these roots be rational? I think so as I'm typing this. I think these are rational: ...
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1answer
51 views

Decide whether the following number is rational

Working needs to be shown $\sqrt{\sqrt{5}+3}+\sqrt{\sqrt{5}-2}$ My guess is to multiply by $\sqrt{\sqrt{5}+3}-\sqrt{\sqrt{5}-2}$ then we have a rational number but is it enough to prove the ...
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5answers
186 views

An Impossible Ratio

I'm facing a bit of a difficulty thinking about the aspect ratio of A4 paper. The beauty of this paper size is that when it is folded in half along the longer side, it becomes A5 paper which has ...
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votes
1answer
98 views

Which irrationals are contained in the Cantor set?

It is well known that the Cantor set is uncountable. Hence it contains irrationals. What are the 'nice' irrationals in the Cantor set. Here, I am expecting irrational numbers in the form of square ...
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0answers
62 views

Roots of polynomials: Vieta's Formula

Let $p_n(t) = c_0 + c_1 t + c_2 t^2 + \ldots + c_n t^n$ with $c_i \in \mathbb{Q}$ and let the roots of $p_n(t) = 0$ be $R = \{r_1, r_2, \ldots r_n \}$. Vieta's formula states that $\sum_{i=1}^n r_i = ...
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3answers
131 views

Irrationality of $n$-th root of positive rationals other than $1$

If $a,b \in \mathbb Z^+ , a \ne b$ then is it true that $\sqrt[a+b]{\dfrac ab}$ is irrational ? This question actually popped up from seeing whether there exists a non-trivial homomorphism from ...
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1answer
43 views

Prove that √m is irrational by showing that n√m is empty

Let m∈N be such that m≠k^2 for all k∈N. Prove that √m is irrational by showing that {n∈N: n√m∈N} must be empty.
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2answers
71 views

Cardinality of set of Dedekind cuts (elementary)

Under the Dedekind construction the irrationals are defined as those cuts $(A,B)$ where $B$ has no least element ($A$ not having a greatest element by definition), for example the $q^2=2$ case. I can ...
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votes
2answers
34 views

$\sqrt{m}$ irrational

Thinking about it, I think I found the following criterion for irrationality of $\sqrt{m}$ if $m$ is a positive integer. Let $p_1^{a_1}\cdots p_k^{a_k}$ be the prime factorization of $m$. Then ...
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2answers
62 views

Difficult denomiator rationalization questions

These are two questions from a competitive exam involving irrationals where I am supposed to simplify it to match one of the given options. QUESTION 1: The value of $$ \frac {2 (\sqrt 2+ ...
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2answers
67 views

Find the values of k for which $\sqrt{1+\frac{k}{n}}$ is irrational.

I would like to find the positive integers $k$ for which $\sqrt{1+\frac{k}{n}}$ is irrational for all $n\in\mathbb{N}$. I was led to this question when I was making up an example for my class, and I ...
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votes
1answer
47 views

Irrationality of Decimal Expansion of Primes

I've heard the proof that this number is irrational is accessible to even a novice to number theory: $\alpha = 0.2 \ 3 \ 5 \ 7 \ 11 \ 13 \ 17 \ldots$ The proof may utilize that a number is ...
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votes
5answers
641 views

How do you solve a logarithm with a non-integer base?

How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example: $$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$ $$\log_{0.5}8 = -3$$ ...
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vote
2answers
60 views

Is such a number necessarily irrational?

Suppose $(q_{n})_{n\in\mathbb{Z}_{\gt 0}}$ is a decreasing sequence of positive rational numbers such that $Q:=\displaystyle{\sum_{n>0}q_{n}}$ is finite. Let's denote by $n_{i}$ and $d_{i}$ the ...
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votes
2answers
137 views

Is $\mathbb{R}\setminus\mathbb{Q}$ a union of countable family of closed sets?

Can we represent set of irrational numbers as union of countable family of closed sets?
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1answer
161 views

How to prove that the problem cannot be solved by the four Arithmetic Operations?

The original prolbem is as in the figure: Suppose the square has unit side length, find the area of blue region. The exact solution is: $$\begin{aligned}S=&\frac{\pi-\sqrt{7}}{4}+2 ...
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2answers
124 views

Does it exist a function that is continuous at every rational point and discontinuous at every irrational point?And vice versa?

Actually there are 2 questions, but they are closely related. Does it exist a function that is: 1. Continuous at every rational point and discontinuous at every irrational point? 2. Continuous at ...
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1answer
60 views

A density question

Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$. Is the set $\{ (2n+1) \theta \bmod 1: n \in \mathbb{N} \}$ dense in $[0,1]$?
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votes
1answer
65 views

Identify irrational basis of $\mathbb{Q}$-vector space

A real sequence $\mathbf{x}=(x_k)_{k\in\mathbb{N}_0}$ is of the form $$ x_k=\alpha r_k,\quad \alpha\in\mathbb{R}\backslash\mathbb{Q},\quad r_k\in\mathbb{Q},\tag{*} $$ if and only if all of its terms ...
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votes
3answers
100 views

Prove that $(√3+2)^{m}$ is not a natural number for all natural numbers $m≥1$

The aim of this question is to show this lemma: Prove that $(√3+2)^{m}$ is not a natural number for all natural numbers $m≥1$.
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0answers
45 views

Sum of 2 different irrational logarithms = Irrational?

I am having some problems proving that the following sum is irrational or rational: $\log_2(3)+\log_3(2)$ = irrational. This is all I've got for now: $\log_2(3)=\frac mn \iff 2^{\frac mn}=3 \iff ...
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1answer
49 views

Nature of the range of $e^x$

Apart from the trivial cases, $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?
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votes
7answers
2k views

How to show that the product of two irrational numbers may be irrational?

Show that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers. All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}\cdot ...
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3answers
121 views

How do irrational numbers lie on the number line?

If we construct a square with side length 1, take its diagonal length : $\sqrt{2}$ However I still don't understand HOW it can lie on the number line. Imagine another irrational number $\pi = ...
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2answers
88 views

How can I prove that the square root of two prime numbers multiplied is non-rational number?

$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?
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votes
2answers
448 views

Irrationality of sum of two logarithms

I try to prove that the number $$\log_2 5 +\log_3 5$$ is irrational. But I have no idea how to do it. Any hints are welcome.
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votes
1answer
75 views

Constraining mathematics to a subset of $\mathbb{R}$

Let's imagine we're only using rational numbers for everything in mathematics. Problems arise quite soon when you try to calculate diagonals of squares or perhaps roots of something like $f(x)=x^2-2$. ...
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votes
1answer
52 views

Does every plane curve contain a rational point?

Does every plane curve contain a rational point? I think the answer is yes, but I can not prove this. Please help. However, if it is possible to build a pathological curve - without rational points, ...
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votes
1answer
89 views

Systematic way to represent any irrational number

I'm wondering if there's a way to symbolically (or is there a more lose constraint?) represent ANY irrational number in a systematic way. You can represent any rational number as two integers and I ...
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vote
6answers
206 views

Proving the irrationality of $\sqrt{5}$

I am working on proving that $\sqrt{5}$ is irrational. I think I have the proof down, there is just one part I am stuck on. How do I prove that $x^2$ is divisible by 5 then x is also divisible by 5. ...
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1answer
64 views

$x$ positive, rational but not an integer. $x^x$ irrational.

Let $x$ be positive, rational, but not an integer. That means $x$ can be written as $\frac{p}{q}$ with $p,q$ coprime, $p,q \neq 0$ and $q \neq 1$. Is $x^x$ always irrational? I think that this has to ...