Questions tagged [rational-numbers]
Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.
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Proof step motivation in proving that set of rationals whose square is less than 2 has no upper bound
I am flaring up my old mathematical hobby and picked up Rudin for reading. My motivation is to read with complete understanding, and thus I am trying to motivate all steps of all propositions laid out....
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Can a function defined on the rationals be continuous?
Suppose we take a function $f:E \to F$, where $E = \mathbb{Q}$, and $F$ is an arbitrary set.
Is it possible for $f$ to be continuous (and thus have a chance at being differentiable)? Or would it ...
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Proving that two distinct rational fractions where all integers are bounded by $N$ are at least $1/N^2$ apart
I recently came across an interesting problem in number theory, namely,
Given distinct integers $a, b, c, d < N$ ( with the condition that $a<b$ and $c<d$ ) , where $N$ is also an integer :
...
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Confusion over whether $1/2$ and $2/4$ should be counted as two different elements of $\mathbb{Q}^{+}$ or not
The instruction for proving that the set of rational numbers, $\mathbb{Q}^{+}$, is countable, is as follows:
First assume each $x\in \mathbb{Q}^{+}$ is in reduced form.
Construct a one-to-one ...
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Sum of three fractions is the sum of their inverses
Question:
What are all the solutions to
$$a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$
for fractions/rational numbers $a,b,c\in\mathbb{Q}$?
I asked a similar question a few weeks ago with perfect ...
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Prove that $\sqrt[n]{a}\in \mathbb{Q}$ [duplicate]
Let $a$ and $b$ be positive rational numbers and $n\ge 2$ be an integer so that $\sqrt[n]{a} + \sqrt[n]{b} \in \mathbb{Q}$. Prove that $\sqrt[n]{a}\in \mathbb{Q}$.
We can obviously write $a=p_1/q_1, ...
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Prove that any rational number can be represented as the square difference of two rational numbers
This proposition was proposed by my deskmate. And I gave a method to work out it.
So I want to communicate with masters of mathematics here.
This is my proof process:
"For $p\in \mathbb{Q}$, ...
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1
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139
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Transcendence of $\ln(a)/\ln(b)$ where $a$ and $b$ are rational numbers.
In this post I proved that ln(3)/ln(2) is transcendental and an immediate corollary is that ln(x)/ln(y) is transcendental where x and y are natural numbers $ x,y \neq 0,1$ if x is odd and y is even or ...
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Number of elements in a continued fraction
When we work with rational numbers, our continued fraction will have a finite number of elements.
Are there ways to estimate the number of elements of a continued fraction when expanding a rational ...
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example of a rational series, conditionally convergent, such that it converges to a rational number
Is there any example of a rational series (of rational terms), conditionally convergent (that is, not absolutely convergent) such that it converges to a rational number?
For example the Leibniz ...
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How many non-homeomorphic subspaces of $\mathbb Q$ are there?
How many non-homeomorphic subspaces of $\mathbb Q$ are there?
$\aleph_1$ ? $\mathfrak c$ ? Is it known?
EDIT.
Since the time I originally posted the question I found out about the result in https://...
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Exact algorithms (e.g. in coding theory, cryptography) using the field of rational numbers
I noticed that most algorithms in coding theory or cryptography are based on the integers and some arithmetic results (e.g. RSA) or on the finite fields (e.g. Elliptic curve cryptography or BCH codes)....
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Examples of famous constants that turned out to be rational
I was just reading through some random articles on the proofs (or lackthere of) of some famous constants being irrational or transcendental. (Such as $ \pi, e$, the Euler-Mascheroni constant etc.)
...
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Show that among the numbers $\sqrt{\frac{pq-2}{3}}$, $\sqrt{\frac{qr-2}{3}}$ and $\sqrt{\frac{pr-2}{ 3}}$ there is at least one irrational one.
question
Let the natural numbers $p$, $q$ and $r$ be greater than $2$. Show that among the numbers $\sqrt{\frac{pq-2}{3}}$, $\sqrt{\frac{qr-2}{3}}$ and $\sqrt{\frac{pr-2}{3}}$ there is at least one ...
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Convergence of rational approximation error series
Question:
For any $x\in\mathbb R$ define the sequence $a_n=\min_{p\in\mathbb Z}|x-p/n|$, does there exist $x\notin\mathbb Z$ such that $\sum_{n=1}^\infty a_n$ converges?
Some thoughts:
If $x\in\...
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Finding accumulation points and isolation points for a given set
I am trying to find $ S' $ for function S:{ r $\in Q$ , $ 0<r\le\sqrt2$ }
Now I believe that $S'={\{0,\sqrt2\}}$ and isolation point is ${\{1}\}$
Now my understanding for this answer is that since $...
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Show that a quarter-circle always contains a point $(x,y)$ with rational $x$ and irrational $y$
Let $r\in\mathbb{R}^{+}.$ Let $\mathbb{D}_r=(0,r)\cap\mathbb{Q}.$ That is, $\mathbb{D}_r$ is the set of all rationals strictly greater than $0,$ and strictly less than $r.$ Let $f_r:\mathbb{D}_r\to\...
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How to define $b^x = \sup B(x)$ for every $x \in \mathbb{R}$ given that $b^r = \sup B(r)$ has been proven for $r \in \mathbb{Q}$
Fix $b > 1$. If $x$ is real, define $B(x)$ to be the set of all numbers $b^t$, where $t$ is rational and $t \le x$. Prove that $b^r = \sup B(r)$ when $r$ is rational. Hence it makes sense to define ...
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Intuition for the Cone Lemma
The Cone Lemma.
If a system of homogenous linear equations with integer coefficients has a
positive real solution, then it also has a positive integer solution.
This is proved in Proofs from THE BOOK,...
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Is $x^6 + bx^3 + b^2$ irreducible?
Let $b\in \mathbb{Q}^*$ be rational number. We factorise $x^9-b^3\in \mathbb{Q}[x]$ and obtain $$x^9-b^3=(x^3-b)(x^6+bx^3+b^2).$$
Is the polynomial $x^6+bx^3+b^2$ irreducible?
If $b=1$ we get a ...
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Why is the Cantor Set not a subset of $\mathbb{Q}$?
I am looking for an example of a specific element of the Cantor set which isn't a rational number, and how it comes about in the set when constructing it, to understand why the Cantor set isn't a ...
- 2,145
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Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$
Let $a$, $b$ and $c$ be nonzero natural numbers. Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$.
My ideas
For those numebrs to ...
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Existence of Bound of Width $\epsilon$ for Non-Cauchy Rational Sequence
Let $\{x_n\}$ be a sequence of Rational numbers: $$\exists \ \ 0 < \epsilon \in \Bbb Q \ \ \exists \ \ N \in \Bbb N: \lvert x_n - x_m \rvert < \epsilon \ \ \forall \ \ n,m \geq N$$
This is not ...
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For continuous $h:\mathbb{R\longrightarrow R}$, prove $h(x)=0$ for x in $\mathbb{Q}$ implies $h(x)=0$ for x in $\mathbb{R}$
The problem statement is as follows: For continuous $h:\mathbb{R\longrightarrow R}$, prove $h(x)=0$ for $x$ in $\mathbb{Q}$ implies $h(x)=0$ for $x$ in $\mathbb{R}$.
My attempt feels a bit awkward and ...
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Rational approximations to multiple real numbers, with shared denominator [duplicate]
To find an optimal rational approximation to a single real number, $x \approx p/q$, one can truncate the continued fraction representation of $x$. Efficient numerical procedures are discussed in ...
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Is this group isomorphic to the real numbers?
I have a locally compact abelian group $G$ with the following properties:
It is connected (therefore divisible) and non-compact;
It admits a $\mathbb{Q}$-vector space structure and for any $g \neq 0$,...
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Is multiplication by a scalar an open map in topological groups?
If n is any non-zero integer and G is a topological abelian group, under what conditions is the continuous homomorphism $g \mapsto ng$ open or weakly open?
If this map happens to be open for all n and ...
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Is there a name for the subset of the rationals where the denominator is coprime with a specific integer?
Considering a natural integer $n$ ($n>1$). Is there a name for the set of all rationals which can be written as $p/q$ with $p$ and $q$ integers and $q$ coprime with $n$ ?
For $n=2$ it would be all ...
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Buffon's Needle Problem
Buffon's Needle Problem
"Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across ...
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Calculating the exact square root of a complex number with rational components [duplicate]
Given a complex number with rational components, I want to check if its square root also has rational components, and if so calculate the value. For example, given $-\frac{119}{225}+\frac{8}{15}i$ I ...
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Name of this modulo operation
I am looking for an operator on $\mathbb{Q}$ that is similar to the modulo operator, but gives me the smallest representative of the class $[0,p]$ instead of $[0,p)$.
Let's call it $\mod'$.
For ...
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Derivatives of real functions which preserve the rationals [duplicate]
I am looking at functions from $\mathbb{R}$ or an open set of it to $\mathbb{R}$ which send rational numbers to rational numbers i.e. $f(\mathbb{Q}) \subseteq \mathbb{Q}$. I will call such functions $...
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Where will irrationals fit on the number line?
Now this might be a very dumb question but this has been bothering me from some days.
Imagine I want to create the real number line and for that I start with the rational numbers. So I start to put ...
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Is a *surface*-line intersection rational if the line passes through another rational point on the surface?
I've seen some interesting geometric interpretations of Diophantine equations before. For instance, if we wish to find Pythagorean triples, we can take $a^2 + b^2 = c^2$, divide through by $c^2$, and ...
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Are all bivariate polynomials of degree < 7 non-injective on rational numbers?
In one of Alon Amit's interesting answers on the Quora website, he mentioned Don Zagier's conjecture that the bivariate polynomial $x^7 + 3y^7$ may be injective on rational numbers, that is, no two ...
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How can I prove a rational number can be expressed as a ratio of two integers, with at least one of them odd?
The proof for $\sqrt{2}$ being irrational relies on the fact that any rational number can be expressed as the ratio of two integers, which means that for every rational number $x$ it is possible to ...
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The rationals and their initial segments [closed]
My question was inspired by an Alon Amit post on Quora recently. The Quora problem posed to
AA was something like, only slightly more confused: how can the set of initial segments of the
rational ...
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Can $\sin(\frac{n\pi}{m})$ with $n,m \in \mathbb{Z}$ always be represented using only algebraic functions?
Can $\sin(\frac{n\pi}{m})$ with $n,m \in \mathbb{Z}$ always be represented using only algebraic functions?
In other words can $sin$ of a rational multiple of $\pi$ always be represented using only ...
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Finding lattice points on ellipses to use Brillhart's factoring method
Theorem 1 (Brillhart 2009). Let $N \gt 1$ be an odd integer expressed in two different ways as
$$N=ma^2+nb^2=mc^2+nd^2,$$
where $a,b,c,d,m,n \in \mathbb{Z}^{+},b \lt d$, and $(ma,nb)=(mc,nd)=1$. Then
$...
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Does the number $\frac{1}{\omega}$ exist?
Edit : Ok, "is not a real number" is a very different statement from "does not exist.". My initial thought is "does not exist.". Sorry, I never know about something like ...
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Summation of the reciprocals of natural numbers which does not have 0 as a digit.
What is the summation of reciprocals according to multiplication of natural numbers which does not have 0 as a digit?
$$
S = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}..+\frac{1}{9}+\frac{1}{11}+...+\frac{1}{...
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Continuity of $f(n):=n,6n\in\mathbb{Z}$
Suppose $f:\frac{1}{6}\mathbb{Z}\to\frac{1}{6}\mathbb{Z}$ is a function defined by $f(n):=n,6n\in\mathbb{Z}$. Is $f$ continuous at $x=0$? A graph of the function somewhat points towards this answer:
...
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Are there three rational numbers $(a,b,c)$ such that $r(a)+r(b)=r(c)$ where $r(q) = \frac{2q(1-q^2)}{(1+q^2)^2}$?
For a given rational number $0<q<1$ let $r(q)$ be
$$
r(q) = \frac{2q(1-q^2)}{(1+q^2)^2}, \quad q \in (0, 1)
$$
Is there a triple $(a, b, c)$ of rational numbers such that
$$
r(a)+r(b)=r(c), \...
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Why $\forall n \in \mathbb{Z}_{\geq 1}$ it is $n^{-1} \leq |n|_*$?
I was reading the proof of Ostrowski's theorem (which BTW is a beauty) and I got stumped here:
[O]ne has $|nr|_∗ \leq n|r|_∗$. [C]hoosing $r=n^{-1}$ shows that for all positive integer $n$, it holds $...
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Rationality of artan(2/pi)
I was messing around on Wolfram Alpha and I saw that the conversion of arctan(2/pi) from radians to degrees was 32.48. Wolfram Alpha usually shows more decimal spaces, so I was wondering where this is ...
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Proof that a repeating decimal has non-repeating digits after decimal iff denuminator has factors of 2 or 5 besides other prime factors
I'm studying a lesson about fractions. It classifies rational numbers into three categories based on their decimal representation.
Terminating Decimal: If a reduced fraction'denuminator has only ...
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Let $a$ and $b$ be rationals. Prove that if $a<b$ then there exists an irrational $x$ such that $a<x<b$.
Let $a$ and $b$ be rationals. Prove that if $a<b$ then there exists an irrational $x$ such that $a<x<b$.
Before anyone starts linking other posts, I have gone through the following posts.
...
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Why is $\mathbb{Q}$ not a $G_\delta$ set? [duplicate]
A $G_\delta$ set is defined as the intersection of a countable family of open sets. If $n \in \mathbb{N}$ and $x_j \in \mathbb{Q}$, $\mathbb{Q}$ can be expressed as $\bigcap\limits_{r=1/n}^{\infty} (\...
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Rational angle dividers of equilateral triangle
let ΔABC be an equilateral triangle. let P be an interior point (strictly interior, not on any border) of ΔABC. let ∠PAB=a1, ∠PBC=a2, ∠PCA=a3, ∠PAC=b1, ∠PBA=b2, ∠PCB=b3. i am interested in finding all ...
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How can two irrational numbers add to a rational number when each doesn't have an irrational part that can simply cancel out? [duplicate]
I think this is asking the same question as here except that I don't quite understand the accepted answer. For instance, I can produce(*) what I think is a counterexample:
$$\sqrt[3]{26+15\sqrt{3}} + ...
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