Numbers not expressible as a ratio of two integers. Examples: $\sqrt{2},\phi,e,\pi,\zeta(3)$. Some of them are algebraic ($\sqrt{2},\phi$) and some transcendental ($e,\pi$).

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3
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2answers
128 views

Simple proof that Pi is irrational - using prime factors of denominator

Simple proof that Pi is irrational Consider the Gregory - Leibniz series for $\pi/4$: $$\pi/4 = 1 - 1/3 + 1/5 \ldots $$ Let $A_n/B_n$ be the irreducible fraction given by partial sum $S_n$ up to the ...
2
votes
3answers
100 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
4
votes
0answers
46 views

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 $c$ ...
0
votes
0answers
30 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.
5
votes
6answers
865 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 ...
0
votes
1answer
20 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?
2
votes
4answers
73 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 ...
0
votes
1answer
31 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 $x\in S$ ...
0
votes
0answers
52 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 ...
2
votes
3answers
79 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 ...
0
votes
1answer
39 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 ...
0
votes
1answer
56 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} ...
1
vote
1answer
84 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 ...
1
vote
2answers
51 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?
6
votes
3answers
94 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?
11
votes
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?
0
votes
1answer
15 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 ...
3
votes
0answers
39 views

Prove that there are infinitely many rational numbers $\frac{m}{n}$ such that accomplish this proposition

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 ...
2
votes
1answer
72 views

1.Why is this proof of “$\sqrt{2}$ is irrational” titled as “Proof by infinite descent”?2. Do I understand it correctly?

I am reading this wikipedea article on the proof of irrationality of $\sqrt{2}$. It uses the principle of infinite descent. I understand it as: We assume $\sqrt{2}=\dfrac pq$, where $p$ and $q$ are ...
5
votes
1answer
132 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 ...
0
votes
0answers
79 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 ...
1
vote
1answer
56 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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votes
3answers
51 views

How to prove that the intersection of $\{p+q \sqrt{2} \mid p,q \in \mathbb{Q}\}$ and $\{r+s \sqrt{3} \mid r,s \in \mathbb{Q}\}$ is $\mathbb Q$? [closed]

Let $S=\{p+q \sqrt{2} \mid p,q \in \mathbb{Q}\}$ and $T=\{r+s \sqrt{3} \mid r,s \in \mathbb{Q}\}$. Prove that $S \cap T = \mathbb{Q}$. We are currently studying proofs by contradiction, but I don't ...
1
vote
1answer
43 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: ...
1
vote
1answer
42 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 ...
2
votes
4answers
61 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 ...
7
votes
1answer
74 views

Irrationals in 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 ...
0
votes
0answers
43 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 = ...
2
votes
3answers
123 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 ...
0
votes
0answers
30 views

Conclusions about addition and multiplication of rational and irrational numbers.

Let $r_1,r_2 \in \mathbb{R}\setminus\mathbb{Q}$, $q\in \mathbb{Q}$ , does the folliwng values are rational, irrational , or we can't decide? $r_1 + q$ $r_1 + r_2$ $r_1 r_2$ $r_1q$ my answers: ...
0
votes
1answer
39 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.
1
vote
2answers
57 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 ...
0
votes
2answers
29 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 ...
1
vote
2answers
47 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+ ...
2
votes
2answers
51 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 ...
2
votes
1answer
40 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 ...
6
votes
5answers
535 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: $$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$ $$\log_{0.5}8 = -3$$ ...
1
vote
2answers
51 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 ...
5
votes
2answers
111 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?
14
votes
1answer
145 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 ...
4
votes
4answers
128 views

Show that $\sqrt[3]{2} + \sqrt[3]{4}$ is irrational

Our professor asked us this to prove that $$ \sqrt[3]{2} + \sqrt[3]{4} \notin \Bbb Q. $$ I know how to prove each one separately that it is irrational, but when it comes to summing two irrational ...
4
votes
3answers
96 views

subtraction of two repeating decimals rationals

When I was looking at the proof that every repeating decimal is rational, I came across this example: $x=5.33333333\ldots$ ($3$ repeat indefinitely) $10x=53.3333333\ldots$ ($3$ repeat indefinitely) ...
0
votes
1answer
37 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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2answers
37 views

Approximation of numbers [closed]

How could we approximate an irrational number by rationals?? Could you give me some hints?? I don`t have any idea how we could approximate them by rationals...
2
votes
1answer
53 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 $[0,1]$?
3
votes
1answer
49 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 ...
2
votes
3answers
83 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 $m≥1$.
1
vote
0answers
41 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 ...
0
votes
1answer
46 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? [closed]

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 $\sqrt{2}^2 = 2$.