For questions on rational numbers, numbers that can be expressed as the quotient or fraction $\frac pq$ of two integers.

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1answer
32 views

Relationship of basis vectors of the complex plane

I am working on learning more about the connection of complex numbers and rotations in the context of rational geometry. Thanks ahead of time for any corrections on my best assertions. Let $B$ ...
-1
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1answer
19 views

Rational vs irrational

What is the difference between a rational and an irrational number? I'm am interested to understand what the difference. Is one like a decimal and one not? Thanks a ton in advance.
4
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2answers
296 views

'Rational' solutions of sine

Do there exist rational numbers $q \in (0,1) \cap \mathbb Q$ such that $$\sin\left(\frac{\pi}{2}q\right) \in \mathbb Q$$ Clearly if $q \in \mathbb Z$, yes. But what about the case $0 < q < 1$? ...
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0answers
21 views

Rationals in an interval $[a,b] \in \Bbb R$

(i) For which real values $a$ and $b$, ($a < b$), is the set $[a,b] \cap \Bbb Q$ open in $(\Bbb Q, d)$, (where $d(x,y)= \lvert x-y \rvert$)? (ii)For which real values $a,b$ is the set $[a,b] \cap ...
2
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1answer
25 views

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational.

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. Take ...
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0answers
47 views

Why rational numbers in stopping times for continuous time processes

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{\ge 0},P)$ be a filtered probability space. Let $X_t \in \mathbb{R}^n$ be a continuous stochastic process adapted to $\mathcal{F}_t$. Let $A \subset ...
3
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1answer
339 views

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gel'fond-Schneider's theorem. I'm interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental ...
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2answers
117 views

I am trying to prove this problem by induction, how can can i prove the following?

I am given $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$ The textbook states that it's equal to the n-th Fibonacci ...
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0answers
43 views

$\{p\in\mathbb Q:p^2<2\}$ having no large element and beyond

In Baby Rudin, the proof of $\{p\in\mathbb Q:p^2<2\}$ having no largest element, a number $q$ larger than $p$ in this set is defined as: $$ q=p-\frac{p^2-2}{p+2}=\frac{2p+2}{p+2} $$ and $$ ...
8
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6answers
2k views

Infinite number of rationals between any two reals.

Let $a$ and $b$ be reals with $a<b$. Show that there are infinitely many rationals $x$ such that $a<x<b$. My plan of action was to assume that $x$ is the smallest such rational and find ...
2
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3answers
55 views

Proving that rational numbers are dense

I am trying to show that for any real number a, there exist infinitely many rational numbers m/n with $ |a - m/n| < 1 /n^{2} $. I've tried to attempt the question by assuming there are finite ...
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2answers
29 views

Problem with the rational root theorem

Consider this polynomial: $f(x)=(2x+5)(x-3)(x+8/3)=0$. Then $f(x)=2x^3+...+(-40)$ Here is a list of all factors of $40$ and $2$: $40$: $±1$, $±2$, $±4$, $±5$, $±8$, $±10$, $±20$ $2$: $±2$, $±1$ ...
4
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1answer
165 views

Equality of positive rational numbers, Part-2

I am reading this answer. I have some doubts which I want to clarify. Question 1. The author defines a rational number $\dfrac ab$ as, $$b\times\left(\dfrac{a}{b}\right) = a$$ He presumes that ...
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4answers
4k views

What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?
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0answers
11 views

What can we say about $\frac{s}{p}$, $\frac{p}{s}$ using these 3 imposed conditions?

What can we say (if anything) about $\frac{s}{p}$ or $\frac{p}{s}$ where $p$ and $s$ are integers greater than $1$ using the following three conditions: $p>s$, $s$ and $p$ are not both divisible ...
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4answers
114 views

Rationality of $e + \pi$

I found just one question similar to this, but it had been edited, so hopefully this isn't asked too often. Given the formulas via infinite sums for expressing $e$ and $\pi$... $$ e = ...
0
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1answer
20 views

Repeated averaging of rational numbers to get zero

I have a set of rational numbers, and the only allowed operation is calculating the mean of a subset and adding it to the set. The goal is to generate zero. I tried brute-forcing this problem with S ...
0
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0answers
19 views

How do I prove that as 2 integers p, s tend to infinity, p/s tends to x?

Forgive me for asking such a broad question, but I really do have very little knowledge on how to do this and it came up in a problem that I have been working on for some time now, so any help would ...
6
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4answers
926 views

Can any two irrational numbers NOT of the form (m+A) and (n-A) be added to produce a rational number?

$m$ and $n$ being rational numbers, A being an irrational number. I was wondering if two irrational numbers when added always yield an irrational number. All the counter-examples I could find were of ...
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2answers
27 views

Is $\mathbb Z$ the only proper sub-domain ( a subring that is an integral domain ) with unity of the ring $\mathbb Q$?

Is $\mathbb Z$ the only proper sub-domain ( a subring that is an integral domain ) with unity of the ring $\mathbb Q$ ? ( I can easily prove that if $D$ is any subring with unity then $\mathbb Z ...
2
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3answers
32 views

How do I solve this rational expression?

I'm stuck on this rational expression. I factored and simplified, by what do I do next? Should I divide x/2x and 8/4? I posted my work below. Thank you!
3
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1answer
60 views

Dedekind's Cuts Lemma

I'm studying Dedekind's Cuts and his construction of Real numbers from the Rational ones. Here we are allowed to use $\Bbb{Q}$ as an ordered field and all all its properties (Archimedean Property, his ...
2
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1answer
43 views

Is it possible to express $\Gamma\!\left(\tfrac{1}{50}\right)$ through values of the $\Gamma$-function at rational points with smaller denominators?

Sometimes it is possible to express a value of the $\Gamma$-function at a rational point through values of the $\Gamma$-function at rational points with smaller denominators, e.g. ...
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0answers
170 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
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2answers
203 views

Is this graph based on rationals familiar?

Has anyone come across a graph like this? The black circles represent rationals in $(0,1)$ and their heights are roughly proportional to the reciprocal of the square of their lowest terms ...
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3answers
33 views

How to evaluate a sum which contains limit variables?

For example: $$\lim_{n\to\infty}\sum_{i=1}^n\frac{n-1}n\frac{1+i(n-1)}n $$ And would the result necessarily be rational, because each term appears to be the multiplication of two rational fractions? ...
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1answer
70 views

Rational summation of irrational numbers

Is the sum of all irrational numbers between any two integer constants rational? I think it should be, because every irrational number should have another irrational with which it would sum to a ...
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4answers
496 views

Sequences of Rationals and Irrationals

Let $(x_n)$ be a sequence that converges to the irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational? Let $(y_n)$ be a sequences that converges to the rational number ...
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3answers
455 views

Irrationals forming rationals

Can we obtain every rational number from the multiplication of two irrational numbers? If not, which ones can we not obtain?
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2answers
85 views

If $a$ is irrational, does there exist a natural number $n$ such that $na$ is rational?

For some irrational $a$, does there exist an $na$ which is contained within the rational numbers for some natural $n$?
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2answers
67 views

Rational and irrational numbers

Consider $x$ a rational number. Let $\epsilon \geq 0$ be the minimal value such that $x + \epsilon$ is irrational, and let also $\gamma > 0$ be the minimal value such that $x+\gamma$ is rational. ...
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2answers
106 views

Given dividend and divisor, can we know the length of nonrepeating part and repeating part?

$13/92=0.14\overline{1304347826086956521739}$ In this example, the length of nonrepeating part is $3$. The length of repeating part (repeating period) is $21$. I collected some properties related to ...
63
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23answers
13k views

Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical. Numerical computations, to my understanding, never deal ...
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0answers
31 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.
56
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3answers
895 views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
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1answer
107 views

Does there exist a unique closest natural number to each rational number?

So here's the question: Prove or disprove: For every $x \in \mathbb{Q}$, there is a unique $n \in \mathbb{N}$ which is the closest natural number to $x$. I know we can define a rational number ...
2
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2answers
56 views

If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms?

Given a rational number $a/b$ expressed in simplest terms (so $GCD(a,b)=1$), I want to raise it to an integer power $n$. I think the result will always automatically be in simplest terms, but it's a ...
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0answers
96 views

Related to $\pi$ and $\tau$ - are these numbers transcendental, irrational, or rational?

Here are three numbers whose decimal expansions are listed in OEIS. Are they transcendental, irrational, or rational numbers? Note: $\tau = 2*\pi$ and the last two values are in radians. A233700. ...
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2answers
275 views

Field containing all square roots of rational numbers

What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
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6answers
897 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 ...
7
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7answers
559 views

Is the number 0.2343434343434.. rational? [duplicate]

Consider the following number: $$x=0.23434343434\dots$$ My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in ...
0
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1answer
29 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?
0
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1answer
33 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
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0answers
67 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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1answer
45 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
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1answer
62 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} ...
16
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8answers
5k views

Proof that every repeating decimal is rational

Wikipedia claims that every repeating decimal represents a rational number. According to the following definition, how can we prove that fact? Definition: A number is rational if it can be written ...
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3answers
327 views

Double limit of $\cos^{2n}(m! \pi x)$ at rationals and irrationals

I stumbled upon this "relation" (is the name correct?): $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is ...
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0answers
43 views

Why is this function surjective?

$f: \mathbb Z \times \mathbb N \rightarrow \mathbb Q$ is surjective? When by definition $\frac pq,\; p, q \in \mathbb Z$ and they can be both positive and negative and $q \neq 0$. Is it possible to ...
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1answer
115 views

Multiplication of positive fractional numbers.

I am reading this answer. We know that multiplication of two positive rational number $\dfrac{a}{b}$ and $\dfrac{r}{s}$ respectivly is defined as follows: $\dfrac{a}{b}\times\dfrac{r}{s} = ...