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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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how to find the excluded value of a fraction that consist of numbers only?

I am trying to solve a worksheet that has questions related to Rational Expressions and one of the questions in that worksheet asks us to simplify and find the excluded value of a rational expression. I got to solve a question that was:- Rational…
Sohaib Farooq
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Help understanding this example of a non-rational number

I watched a video https://www.youtube.com/watch?v=isbt-7DQBy0 in which the instructor gives an example of a non-rational number. He says, if the square root of 2 would be a rational number, it could be expressed as the fraction of two…
BMBM
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Consider $a,b\in\mathbb{R}$ where $a < b$. Use the Denseness of $\mathbb{Q}$ to show there are infinitely many rationals between a and b.

The Denseness of $\mathbb{Q}$ says that if $a,b\in\mathbb{R}$ and $a < b$, then there exists a rational $r\in\mathbb{Q}$ such that $a < r < b$. My question is, how do I show that there is infinitely many of something? I know that I could divide r…
Garrett
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Prove $P \implies Q$ where $P$ and $Q$ are statements about rational numbers

My goal is to prove the following statement $\left(P\implies Q\right)$ in order to use it as a lemma in another proof. Given, $$P = \left(z\in\Bbb Q \implies \exists \,p,q\in\Bbb Z:z\,=\,\frac pq\wedge q \ne 0 \right)$$ consider the…
user409521
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Marking a repeating decimal when it's written with a set number of places

I'm typesetting some film related material that contains a list of common aspect ratios: 1.33 1.66 1.77 1.85 2.35 And this is exactly how they are usually displayed, using two decimals. Now, the ratios 1.33, 1.66 and 1.77 all have a single…
typo
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rational number word problem

I don't want algebraic solution to this question as i already have it.. as i have to teach it to a student with no algebra knowledge. There are 42 pupils in a class. 3/4 of the boys and 2/3 of the girls travel to school by bus. The total number of…
Tasawar Hussain
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Proof about rational neighbors

Two rational numbers $\frac{a}{b}$ < $\frac{c}{d}$ will be called neighbors if $\frac{c}{d}$ - $\frac{a}{b}$ = $\frac{bc-ad}{bd}$ = $\frac{1}{bd}$. Suppose $\frac{a}{b}$ and $\frac{c}{d}$ are neighbors in this sense and $\frac{m}{n}$ is a rational…
AndroidFish
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Looking for a simple proof of why you can't mathematically tune a piano

https://www.youtube.com/watch?v=1Hqm0dYKUx4 Video states that a corollary of the Rational Root Theorem is that $\left(\frac{a}{b}\right)^n != 2$ for integers $a,b,n$, where $n \gt 1$. I'm simply looking for a proof by contradiction of this. I…
Interested
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How to prove that $y = 0,273273273...,$ is a rational number?

How to prove that $$y = 0.273273273...$$ is a rational number? I don't have any experience with proofs... Can I get your help and your advice?
Always learning
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What decimal is between 0.5 and 0.625

I would really appreciate some help with this. I have been literally stumped with it for an hour. So if you know the answer please comment below! Thank you for your time:)
Sally
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using long division to find the oblique asymptote of rational function

To find the oblique asymptote of a rational function, the book I'm reading says to divide the denominator of a fraction into the numerator. The example rational function it gives is $$\frac{x^2 - 9} {x + 2}.$$ The result of the long division it says…
Michael
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Rational number with rational exponent becomes rational

I'm looking for a proof to show when $p^q$ for $p,q \in \mathbb{Q}$ is again in $\mathbb{Q}$, without factoring. I'm not sure, if it's possible, given these two numbers to say if the result is again rational and if so, calculate the result…
Robert Eisele
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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 = {7, -4} but failed. Does this problem have a…
Toby Kelsey
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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 $m$ digits and the denominator is to be at most $n$…
Trogdor
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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 find two integers $p$ and $q$ so that $x =…
Mat
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