# 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 by Induction Question including Rational Numbers

I just recently covered 'rational numbers' in class and was assigned the following question to solve using induction for n, so that for all $q \in \mathbb{Q}$ \ {1}:...
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### I don't understand why this axiom makes the difference between a completely ordered field and an ordered field

I've been reading about ordered fields and completely ordered fields, and I'm stuck on the difference between the rational and real numbers that makes the rational numbers an ordered field and the ...
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### A course of Pure mathematics incorrect

I thought I'd go through 'A Course Of Pure Mathematics' by G H Hardy And I came to this part: If the reader will mark off on the line all the points corresponding to the rational numbers whose ...
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### Show that if $x$ has a terminating decimal expansion then $x=p/q$ for integers $p,q$ where the only prime factors of q are 2's and 5's.

I've proven the converse statement but don't know where to start for this statement. Would you suggest doing a proof by contradiction?
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### The set {${m^m}/{n^n}: m,n\in \mathbb{N}$} density in ${\mathbb{Q}_+}$

I missed this topic (density) , so need help. Is the set {${m^m}/{n^n}: m,n\in \mathbb{N}$} everywhere dense in positive rational numbers set? I think maybe first of all here we must try with, for ...
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### Evaluating $\sum_{(a,b,c)\in T}\frac{2^a}{3^b 5^c}$, for $T$ the set of all positive integer triples $(a,b,c)$ forming a triangle

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a$, $b$, $c$. Express $$\sum\limits_{(a,b,c)\,\in\,T}\;\frac{2^a}{3^b 5^c}$$ as ...
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### Abelian Group of Rationals over addition - axiomatic or derived? [closed]

Sorry if this is a daft question, but if we consider the set of rationals under the 'addition' operator we can form an Abelian Group. I'm curious: in that situation, is the definition of addition (...
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### Show that rational numbers have the Archimedean property

By Archimedean property I mean: For any positive rational numbers $x = \frac{a}b$ and $y = \frac{c}d$, there is an integer $n$ such that $nx > y$, namely, $nx \equiv (x+x+ \ldots+x)$ with $n$ ...
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### $\mathbb{Z}\leq G \leq \mathbb{Q}$ and $\varphi:G\rightarrow H$ a map which behaves nicely on fractions.

Let $(H,+)$ be any abelian group, and let $\mathbb{Z}\leq G\leq \mathbb{Q}$ a subgroup of the rational numbers with respect to addition. Let $\varphi:G\rightarrow H$ be a map with the following ...
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### A family of subspaces of $\mathbb R^4$ which does not intersect non-trivially a rational subspace

This question is somehow linked to this previous question I asked earlier. Let's consider the following family $F$ of subspaces of dimension $2$ of $\mathbb R^4$. A subspace $A$ is in $F$ if it ...
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### Is this particular subspace of $\mathbb R^4$ intersecting non-trivially a rational subspace?

Let's consider $A$ the subspace of $\mathbb R^4$ given by $A=\mathrm{Span}(Y_1,Y_2)$ where $$\begin{cases} Y_1=(0,1,\sqrt 2,\sqrt 3), \\ Y_2=(1,0,\sqrt 3,\sqrt 2).\end{cases}.$$ I was wondering ...
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### Are rational points dense on every circle in the coordinate plane?

Are rational points dense on every circle in the coordinate plane? First thing first I know that rational points are dense on the unit circle. However, I am not so sure how to show that rational ...
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### Does the supremum of the set $S=\{x\in\mathbb{Q}: x>2\}\subset \mathbb{R}$ exist?

Does the supremum of the set $S=\{x\in\mathbb{Q}: x>2\}\subset \mathbb{R}$ exist? If not, prove it. I don't think the supremum exist, but I don't know how to prove it.
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### Rational numbers in irrational bases

If you take the base-$b$ expansion of a rational number where $b$ is irrational, do you get a non-terminating sequence of digits (assuming you pick the right(?) digits)? More informally, do rational ...
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What is the Minkowski sum of Q to itself, where Q is the set of all rational numbers? I can't find a way to solve this with freshman knowledge. Thank you in advance.
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### Proof by contradiction - am I correct?

I am trying to learn some discrete mathematics alongside my arts course to try and expand my knowledge. I have this question: Prove that if $x$ is irrational then $\frac{x+1}{x-1}$ is irrational. My ...
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### Irrationality of $r\sqrt{5}$ for rational number $r$? [duplicate]

As $5$ is a prime number, thus $\sqrt{5}$ is an irrational number. Now I am thinking about how to prove - If $r$ is a rational number, then how do we prove $r\sqrt{5}$ is an irrational number? I ...
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### When is the denominator of the sum of two positive fractions not divisible by the denominator of either numbers?

Let $r_1 < r_2$ be two positive rational fractions, both $> 1$ and both in their lowest terms i.e. the numerator and denominator of each fraction have no common factors. If $r_3 = r_1 + r_2$ is ...
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### Checking for rational numbers with small denominators in an interval

I have been given two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, and I have been told that there is a unique rational number $\frac{p}{q}$ satisfying $|q| \leq n$ (where $n$ is some known ...
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### Two whole numbers n and k . Print k decimal digits of 1 / n .

I have to print these decimal numbers in C++ . But first i need to understand this question mathematically .
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### Is there a well-behaved finitely additive “measure” on $\mathbb{Q }$?

There exists no nontrivial measure on the set of rational numbers for which the measure of singletons is zero. That’s because the rational numbers are countable, so any set of rational numbers is a ...
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### Is $[0,1] ∩ Q$ open or closed in real numbers?

According to https://en.wikipedia.org/wiki/Closed_set, " $(0,1) ∩ Q^c$ is not closed in the real numbers". Why is it so? I think that its complement is $(-∞,0)∪ (1,∞) ∪ Q^c$ is open and all points ...
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### Vector field in $\Bbb Q^2$ and $\Bbb R^2$

I'd like to define a weak vector field in $\Bbb R^2$ that is tangential to a family of sine curves at each point. I define the family of sine curves as: $A_n\sin(x)$; $n=1,2,3,...$ and $A_n$ is a ...
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### $18a$ and $25a$ both integers, then so is $a$

Let $a\in \mathbb{Q}$ such that $18a$ and $25a$ are integers, then we wish to prove that $a$ must be an integer itself. What that means is that $a=\frac{p}{1}$ where $p \in \mathbb{Z}$. What we do ...