# 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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### 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 ...
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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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### 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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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 $... 0 votes 0 answers 61 views ### Rationality of artan(2/pi) I was messing around on Wolfram Alpha and I saw that the conversion of arctan(2/pi) from radians to degrees was 32.48. Wolfram Alpha usually shows more decimal spaces, so I was wondering where this is ... 1 vote 2 answers 204 views ### Proof that a repeating decimal has non-repeating digits after decimal iff denuminator has factors of 2 or 5 besides other prime factors I'm studying a lesson about fractions. It classifies rational numbers into three categories based on their decimal representation. Terminating Decimal: If a reduced fraction'denuminator has only ... 2 votes 1 answer 199 views ### Let$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$. Before anyone starts linking other posts, I have gone through the following posts. ... 0 votes 2 answers 139 views ### Why is$\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} (\...
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{26+15\sqrt{3}} + ...