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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 if $a$ is a rational number and $a^2$ is an integer then $a$ is an integer.

Question on a proof's review: Proof by contradiction: Suppose $a$ is not an integer. Then $a=p/q$ where $p$ and $q$ are coprime, $q$ is not 0, and $q$ is not 1. Then $a^2 = p^2/q^2$. This is as far as I've gotten, my t.a. says I'm on the right…
user122661
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Prove that a number $a$ is rational if and only if there exists a positive integer $k$ such that $[ka]=ka$; likewise for $[k! a]= k! a$

How can I go about proving the following problem: Prove that a number $a$ is rational if and only if there exists a positive integer $k$ such that $[ka]=ka$. Prove that a number $a$ is rational if and only if there exists $k$ such that $[k! a]= k!…
phoenix
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Prove $\forall u,v,x,y,z,w \in \mathbf{R}^+, \frac{u}{v} < \frac{x}{y} \wedge \frac{x}{y} < \frac{z}{w} \implies \frac{u + z}{v+w} < \frac{z}{w}$

This is a question from a past exam that I can't seem to figure out. Any tips or hints? Prove $$\forall u,v,x,y,z,w \in \mathbf{R}^+, \frac{u}{v} < \frac{x}{y} \wedge \frac{x}{y} < \frac{z}{w} \implies \frac{u + z}{v+w} < \frac{z}{w}$$ EDIT: My…
jmazz
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find $u < r$, $v < s$ such that $p< uv < rs$

Let $p < rs$ where both $p, r, s$ are positive rational numbers. I want to find rationals $u < r$, $v
Kim Seong Hyeon
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Is a Whole Number A Rational Number

Is a Whole Number part of A Rational Number or a whole number??
Alfred
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How to prove that $X$ is not rational $X = Z - \pi \times Y$ where $Y$ is rational , $Z$ is an integer, $\times$ means multiplication

Excuse me for a silly question like this. I am 60 years old retired engineer and want to learn some basic math I di did not learn earlier. I know an example where $(x + y\pi)$ can be an integer, where $y$ is rational , * means multiplication But…
Shaldar2021
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Assume both r and s are rational then $\frac{r}{s}\in\mathbb{Q}$

Currently done with showing for both r+s and r $\times s$ rational. Now stuck with the following part; If we know that r and s are both rationals what could we say about $\frac{r}{s}$ Assume both r and s are rationals. Then we would have by…
Mark
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Prove that $a$ and $b$ are rational numbers

If $a+b$, $a^2+b$ and $b^2+a$ are rational numbers and $a+b\neq 1$ then $a$ and $b$ are rational. I try, sum the expresions but I only got that $a^2+b^2$ and $ab$ are rational. Any suggestion?
Or Esc
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Square root of 6 proof rationality

I was proving $\sqrt 6 \notin \Bbb Q$, by assuming its negation and stating that: $\exists (p,q) \in \Bbb Z \times \Bbb Z^*/ \gcd(p,q) = 1$, and $\sqrt 6 = (p/q)$. $\implies p^2 = 2 \times 3q^2 \implies \exists k \in \Bbb Z; p = 2k \implies 2k^2 =…
Papa
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Representing zero as a rational number

How to represent 0 as rational number? $0/0$ is not legitimate, $0/\text{const}$ should be good enough, but what is the right value of const? $0/1$ works for a lot of computational cases, but only on intuitive.
SkyFox
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is "commensurability" as simple as saying a and b are rational?

I am trying to better understand commensurability. Wikipedia says: two non-zero real numbers a and b are said to be commensurable if $\frac{a}{b}$ is a rational number. Richard Courant in Introduction to Calculus and Analysis says: Two…
Stephen
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Solving Rational Equation

I got this question and I'm stuck on one of the steps. And this is where I'm stuck.I'm not sure what to do next. Do I multiply everything by 2(a-2)(a+2) over 1 or something entirely different?
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What actually "forces" us to define the operations on $\mathbb{Q}$ the way they are?

Given the integers, we have the usual operations which behave the way they are expected to behave. Now suppose we want to create $\mathbb{Q}$ which are pairs $\cfrac{a}{b\neq 0}$. We want that $\mathbb{Q}$ have a copy of $\mathbb{Z}$ that behaves…
Red Banana
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Rational Number Identity in Elliptic Functions According To Eisenstein and Kroneckor

I am reading "Elliptic Functions According to Eisenstein and Kronecker" and I am struggling with a particular formula derivation: $$ \frac{1}{p^mq^n} = \sum_{h=0}^{m-1} \frac{n(n+1)...(n+h-1)}{h!p^{m-h}r^{n+h}} + \sum_{k=0}^{n-1}…
Hugh Entwistle
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If $a$ and $b$ are positive rational numbers with $a < b$, show that $\frac{1}{a} >\frac {1}{b}$

Since both $a,b\in \mathbb{Q}^+$ and $a
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