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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".

2231 questions
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Position or rank of an arbitrary rational number

Rational numbers are countable as shown by the usual table here: https://aminsaied.wordpress.com/2012/05/21/diagonal-arguments/ So, counting in the zig-zag manner as shown in the table, $1/1$ is the first rational, $3/2$ is the eighth, $1/4$ is the…
Peter4075
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Finding the next rational number

A rational number is one that can be written as $a/b$ where $a$ and $b$ are integers, $b\gt0$ ($a$ can take care of negative rationals), and I suppose $\gcd(a,b) = 1$. Given some $n\in\mathbb{Q}$ where $n=a/b$, what is the next rational number? At…
pushkin
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Online math question didn't do a square of a square root properly?

I'm currently taking an online college math course, and I recently came across something that I can't make any sense out of. $(\frac{5x \cdot \sqrt{3}}{6})^2 = \frac{25x^2}{12}$ It looks like the problem is just completely throwing away the…
MrHost56
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If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms?

Given a rational number $a/b$ expressed in simplest terms (so $GCD(a,b)=1$), I want to raise it to an integer power $n$. I think the result will always automatically be in simplest terms, but it's a long time since I was doing maths regularly, so…
user510
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Finding the simplest rational in a closed interval

Given a closed interval [a,b], how would you find the "simplest rational", p/q, contained in that interval. By simplest, I mean the rational with the smallest denominator q. You may, if you wish assume that the interval is very small. We can already…
Scot
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question about the proof about the square root of natural numbers

Could someone please help me to prove that for $t \in \mathbb{N}$ , $\sqrt{t} \in \mathbb{Q} $ if only if $\sqrt{t} \in \mathbb{N}$
user131215
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Rational doubt ( doubt in rational number)

If there is a prime number x, if we reciprocate it we will get 1/x. Reciprocal of prime number will be a rational number , Except 1/2 and 1/5 , every number which is reciprocal of prime number is a recurring non terminating decimal…
user136567
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Proof by contradiction problem on rational numbers

Using proofs by contradiction, show that there is no smallest negative rational number and no largest positive rational number. Assume that there is a smallest negative rational number. Therefore, the number is of the form $r = - \frac{p}{q}$, where…
nightmarish
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What is the name of the following theorem used to prove that rationals are not complete?

if $\lambda^2\lt2$ then there exist $\epsilon\in\mathbb{Q}, \epsilon\gt0$ such that $(\lambda+\epsilon)^2\lt2$I've seen it in a proof showing that the set of rational numbers is not complete. It is mentioned also in this answer :…
rik
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Prove that $\sqrt 2 + \sqrt[3]2$ is not rational

How do I prove that the following is not rational? $$x=\sqrt 2 + \sqrt[3]2$$ To prove a simpler case like $\sqrt{2}=a/b$, I can raise both sides to the power of 2 and get $a^2=2b^2$, therefore both $a$ and $b$ must be even numbers which can't be…
amin
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Is there a method to figure out how many decimal places there are for a recurring decimal?

For example, like for 1/7 = 0.14285714285. 142857 is repeated in this recurring decimal, and those 6 decimal places are repeated. So is there a method to figure out how many decimal places there are in certain recurring decimals?
Sean Voon
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Wrong Answer - Rewrite Rational Number as a Fraction.

This number 2.962962 can be rational $$x=2.962962$$ $$10x=29.62962$$ $$100x=296.2962$$ $$1000x=2962.962$$ $$1000x-10x=\frac{990x}{990}=\frac{2933}{990}$$ why is this wrong? That way of getting the answer is how I was said to do…
user1838781
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Ordering of rationals

Let $(\mathbb{Q},<)$ be the usual ordering of rationals. Show that there is a family $\mathcal F$ of subsets of $\mathbb{Q}$ such that $|\mathcal F|=2^\omega$ and for every $A, B \in \mathcal F, (A,<)\ncong (B,<).$ I know that the question is asking…
taupi
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Are Rationals constructed from infinite Naturals valid?

I can construct a Rational like $3/4$. And then I can construct anther one like $31/41$, and then another like $311/411$. I can envisage a Rational whose numerator is $31111111...$ and denominator is $41111111...$, where the $1$s continue…
Al.
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Prove $x \in \mathbb{Q}$ and $y \notin \mathbb{Q} \implies (x + y) \notin \mathbb{Q}$

Looking for tips to prove the homework $\forall x,y \in \mathbb{R}, x \in \mathbb{Q} \land y \notin \mathbb{Q} \implies (x + y) \notin \mathbb{Q}$ Can I assume the hypothesis and to yield a contradiction assume that $(x + y)$ is rational, or rather,…
Leonardo
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