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

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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 ...
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1answer
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$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ 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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46 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 ...
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42 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 $$ ...
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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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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 $2$: $40$: $±1$, $±2$, $±4$, $±5$, $±8$, $±10$, $±20$ $2$: $±2$, $±1$ ...
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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
113 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 = ...
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1answer
19 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 ...
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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 ...
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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 ...
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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!
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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 ...
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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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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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453 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
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 ...
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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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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.
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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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3answers
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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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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
106 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 ...
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1answer
27 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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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 $x\in S$ ...
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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 ...
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1answer
61 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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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
163 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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3answers
104 views

Proving Floor and Ceiling of a Rational Number

Suppose x,y $ \in \mathbb{Z}^+ $ Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$ I was considering using the definition of floor and ceiling to prove this. But this does not seem like a ...
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1answer
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$(-3)^{3/2} \neq (-3)^{6/4}$

$(-3)^{\frac{3}{2}}=-3\sqrt{3}i$ $(-3)^{\frac{6}{4}}=\sqrt{27}$ (not the same thing). What's the deal? It's interesting because people work with fractional exponents all the time and I've never ...
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2answers
83 views

An isomorphic map from natural numbers to positive rational numbers that preserves addition, multiplication and order [closed]

Since $\mathbb{Q}^{+}$ is countable, there is a bijection between $\mathbb{Q}^{+}$ and $\mathbb{N}$ (0 included). Then the question now is, can we go further by constructing an isomorphic map between ...
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3answers
50 views

Finding the simplest rational in a closed interval

Given a closed interval [a,b], how would you find the "simplest rational", p/q, contained in that interval. By simplest, I mean the rational with the smallest denominator q. You may, if you wish ...
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2answers
42 views

Is the sum of rational exponentials a rational exponential?.

Prove or disprove that $\forall a,b \in \mathbb{Q}^+$ and $ \forall p,q \in \mathbb{Q}$ there exists $c \in \mathbb{Q}^+$ and $r \in \mathbb{Q}$ such that: $$ a^p+b^q=c^r $$
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44 views

Path connectedness of the set of points $(x,y)$ where $x$ is rational or $y$ is rational [duplicate]

Prove that $X=\{(x,y) :x\text{ is rational or }y\text{ is rational}\}$ is path connected. So for every $(x,y)$ in $X$, I need to find a continuous function $f$ on $[a,b]$ such that $f(a)=x$ and ...
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36 views

How many are there triangles with different rational sides, rational area, bisectrixes and 1 rational median?

I've been searching triangles with all elements being rational numbers. However, I've found somewhere on Internet proof that it's not possible. Then, I was searching triangles with maximal possible ...
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45 views

Does the equation $\tan(x)=y$ have any non-zero rational solution?

Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) ...
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51 views

Quadratic Equations - One rational solution?

I have a question that I am working on: Which of the following will give one rational solution? 4x^2 = 9 4x^2 - 12x = -9 x^2 = 5 x^2 - 2x + 14 = 0 2x^2 = x I am ...
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How to prove that the function $f(x) = 2 \left \lfloor x \right \rfloor - x$ is one to one for rational $x$?

How to prove that the function $f(x) = 2 \left \lfloor x \right \rfloor - x$ is one to one for rational $x$? I believe that I will have to somehow use the fact that the $\left \lfloor x \right ...
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2answers
59 views

$a_1^k+a_2^k+\ldots+a_n^k$ integer implies all integers?

Let $n$ be a positive integer, and let $a_1,\ldots,a_n$ be rational numbers. Suppose that $a_1^k+a_2^k+\ldots+a_n^k$ is an integer for all positive integers $k$. Is it true that $a_1,a_2,\ldots,a_n$ ...
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Given conditions on the fraction, can we find a 'best rational approximation'

Just something I thought of and I'm curious about. Say I tell you I want to approximate $\pi$ using a rational number. However, I am going to tell you that the numerator is to be at most $m$ ...
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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 ...
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1answer
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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
38 views

Is the set $ \{(p_1,p_2,\dots, p_n):p_i\in \mathbb Q\}$ connected?

Let $X=\{(p_1,p_2,\dots, p_n):p_i\in \mathbb Q\}$. Is $X $ connected or disconnected? My attempt:$X$ is connected iff any two points of $X$ are contained in a connected subset of $X$. This ...
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Properties of digit functions for numbers in $[0,1]$

Consider a function $g(n): \mathbb N \to \{0,1,2,3,4,5,6,7,8,9\}$, ie. $g$ maps the natural numbers to natural numbers between $0$ and $9$. Then, no matter what $g(n), \ n\in \mathbb N$ is, the sum ...
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3answers
124 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
58 views

$\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are rational

Are there infinitely many pairs of different positive rational numbers $x,y$ such that $\sqrt{x^2+y^3}$ and $\sqrt{x^3+y^2}$ are rational? Consider such a pair. Then we have $x^2+y^3=a^2$ and ...