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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.

A rational number is any number that can be expressed as the quotient or fraction $\frac pq$ of two integers, with the denominator $q$ not equal to zero. Since $q$ may be 1, every integer is a rational number. The set of all rational numbers is usually denoted by $\Bbb Q$; it was thus named in 1895 by Peano after quoziente, Italian for "quotient".

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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 $p\in \mathbb{Q}$, choose $k\in \mathbb{Q}$ (and $k \neq 0$) construct $m, n$…
Absolute Value
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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 the rationals with an odd denominator, for $n=3$ all…
Anne Aunyme
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Is $\sum_{k=1}^{\infty} 1/k^2 = \pi^2/6$ a valid example for the incompleteness of rational numbers?

Consider the sequence of partial sums $s_n = \sum_{k=1}^{n} \frac{1}{k^2}$. As a finite sum of rational numbers, the $s_n$ itsself are rational. However, since $\pi$ is transcendental, $\pi^2$ is an irrational number. We can show that $s_n$ is…
some_user
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What it means? How to solved it?

For any positive rational number $u$, let us agree to call the numbers $u + 1$ and $u/(u+1)$ the children of $u$. Show that every rational number is the descendant of $1$ in a unique way.
SAMQ
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Prove that if two rational numbers are equal, then they are proportional.

If $(m_1, n_1)$ and $(m_2, n_2)$ are two equal rational numbers, i.e. $m_1*n_1^{-1} = m_2 * n_2^{-1}$, so $m_1*n_2 = m_2*n_1$. And if $m_1$ and $n_1$ are coprime, prove that $n_2 * n_1^{-1} = m_2 * m_1^{-1} = k \in \mathbb Z$. The main problem is…
Nikita Tkachuk
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Countability of the rationals

I have read some of the proofs about countability of the rationals and I am okay with those proofs. But I find a way to get into trouble by doing it like this - The set of all rationals, of the form $\frac{a}{b}$, can be expressed as a set of…
ahron
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To whom the properties belong?

There are many properties like the closure property, commutative property, associative property, distributive property. Now my question is are these the properties of numbers or operations. Should I say the closure property of integers or…
Mohd Saad
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Can somebody construct multiplication of rational numbers?

Imagine you being an ancient person, developing a theory of rational numbers from scratch, and suppose, you've discovered all known properties of integers (you've extended, previously found by you, natural numbers for operating with a notion of…
Maksym Khomiak
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recounting the rationals typo

In recounting the rationals. Is this statement a typo: Finally, the rightmost vertex of each row has denominator 1, as does the leftmost vertex of the next row, proving the claim. Did they mean "the numerator of the leftmost vertex of the next…
sniperking
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Misconception about basic mixed fraction

We know, $$3 \frac12=3+\frac12$$ Then, if we have $$3 \frac12 ÷ 3 \frac12$$ It means: $a)\,\frac72 ÷ \frac72=1$ or $b)\, 3+\frac12 ÷ 3+\frac12 =\frac{7}2$ or $c)\, 3 + 1 ÷ 2 ÷ 3 + 1 ÷ 2=\frac{11}3$ Which one is true? Sorry, maybe it appears on…
user516076
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Finding minimal period of rational

I'm trying to find the decimal representation minimal period of $1/n$ where $n$ is an integer. I'll clarify colloquially because I'm very noob with math terms: $$1/3 = 0,(3)$$ $$DP(3) = 1$$ $$1/7 = 0.(142857)$$ $$DP(7) = 6$$ After a long search on…
ichigolas
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Inclusion of $Z$ in $\mathbb{Q}$

When constructing $\mathbb{Q}$ as equivalence classes containing pairs of integers, a natural inclusion of $\mathbb{Z}$ in $\mathbb{Q}$ arises: $$f: \mathbb{Z} \to \mathbb{Q}, \; n \to \frac{n}{1}. $$ This mapping is not surjective, but it is…
user465188
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For a rational number $x \in [-a/4, a/4)$ with a positive integer $a$, is it possible to separate it into its integral and fractional parts?

For a rational number $x \in [-a/4, a/4)$ for a positive integer $a$, is it possible to separate it into its integral and fractional parts? Namely, can we represent $x$ as $x = b + c$ where $b \in \mathbb{Z}_a, c \in [-1/4, 1/4)$?
user9414424
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Ratio and Proportion - IV

If $a,b,c,d$ are continued proportion : Prove that : $(\frac{a-b}{c}+\frac{a-c}{b})^2-(\frac{d-b}{c}+\frac{d-c}{b})^2=(a-d)(\frac{1}{c^2}-\frac{1}{b^2})^2$ After solving LH.S I got : $\frac{2(a-d)}{(bc)^2}$ But after solving R.H.S I am getting…
Sachin Sharmaa
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