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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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Show that $p/q$ is the$ [(1/2)(p+q-1)(p+q-2)+q]$th term of the series

I am attempting to prove that given a series of rational numbers $p/q$ as presented below: $$ 1/1,\; 2/1,\; 1/2,\; 3/1,\; 2/2,\; 1/3,\; 4/1,\; 3/2,\; 2/3,\; 1/4,\; \ldots $$ That $p/q$ is the $[(1/2)(p+q-1)(p+q-2)+q]$th term of the series. I…
GovEcon
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How to find out the number of repeating digits of a rational number in decimal form?

Upon dividing two integers, I would like to programmatically predict the number of decimal places that repeat after the decimal point. For example in $\frac{1}{3}=0.\overline{3}$, I want to know that the number of repeating digits is $1$. In…
BenMorel
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Determine if $(p/q)^{a/b}$ is rational

I know, in general, that it isn't true. ${\frac{2}{1}}^{1/2}$ is irrational. I'm only interested in this where $\frac{p}{q}$ and $\frac{a}{b}$ are positive, but to make this even simpler, lets just say that $a,b,p,q \in \mathbb{N}$. I'm curious to…
Andy Shulman
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Using a given identity to solve for the value of an expression

This problem caught my eye in the book yesterday. Till now I still get stuck. Here it is: If $$\frac{x}{x^2+1}=\frac{1}{3},$$ what is the value of $$\frac{x^3}{x^6+x^5+x^4+x^3+x^2+x+1}?$$ The denominator is a cyclotomic polynomial which can be…
keith_here
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Summation of the reciprocals of natural numbers which does not have 0 as a digit.

What is the summation of reciprocals according to multiplication of natural numbers which does not have 0 as a digit? $$ S = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}..+\frac{1}{9}+\frac{1}{11}+...+\frac{1}{99}+\frac{1}{111}... $$ I try to think first all…
aileia
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Prove that $2\times3 = 6$ using Dededkind cuts

I'm reading Classic Set Theory by Goldrei, and in Exercise 2.10, after defining real multiplication using Dedekind cuts, I'm asked to prove: Show that $2 +_{\mathbb{R}} 3 = 5$ and $2 \cdot_{\mathbb{R}}3 = 6$. The sum is easy, but I can't do it for…
Franco Victorio
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Naive height of zero

The naive height function of a rational number $x=\cfrac{m}{n}$ (in lowest terms) is defined as $$H(x) = H\left(\frac{m}{n}\right) = \max\{|m|, |n|\} $$ However, $0$ can be denoted by $\frac{0}{1}, \frac{0}{2}, ...$ Then what's the value of…
user831162
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bijection between $\mathbb{Q}$ and $\mathbb{N}$ that preserve the order.

I know there is a bijection between $\mathbb{Q}$ and $\mathbb{N}$. But is there a bijection $\mathbb{Q}\xrightarrow{f}\mathbb{N}$ that preserves the order? Intuitively I think this is not possible. What I would think of is enumerating the rationals…
roi_saumon
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Infinite sum of rationals is irrational

a) Let $x$ be a number between $0$ and $1$. Let $a_1$ be the smallest positive integer such that $x_1=x-a_1^{-1}\geq 0$, let $a_2$ be the smallest positive integer such that $x_2=x_1-a_2^{-1}\geq 0$, etc. Show that this leads to a finite…
ahahahaaa
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Is a subset of $\mathbb{Q}\times\mathbb{Q}$ that all variants of an exponentiation equation have answers in it, infinite?

Note that we have: $$A=\{(a,b)\in\mathbb{Q}\times\mathbb{Q}~|~\text{Both equations}~a+x=b, b+y=a~\text{have answers in }~\mathbb{Q}\}=\mathbb{Q}\times\mathbb{Q}$$ $$B=\{(a,b)\in\mathbb{Q}\times\mathbb{Q}~|~\text{Both equations}~a\times x=b, b\times…
user200478
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Is a complex fraction considered part of the rationals?

I have always been taught that $\mathbb{Q}=\{ \frac{a}{b}|\,\,a,b\in \mathbb{Z},\, \,b\neq0\}$. Is this definition of the rationals limited? Could it also be true that a complex fraction, i.e. $\frac{\frac{a}{b}}{\frac{c}{d}}$ is also a rational…
Michael
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Length of a rationals period in base $b$

Okay working in base $b$ we are given a fraction of form $\frac{p}{q}$ with $p$ and $q$ coprime. We also assume that $b$ and $q$ are coprime so $\frac{p}{q}$ is purely periodic in base $b$. The question I have is what can I say about this period.…
ruler501
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Converting Repeating Decimal Numbers to Fractions

Is it possible to write any decimal number, with a repeating decimal part, and be able to convert it into the form $\frac nd$ (where both $n$ and $d$ are natural numbers)? I know rational numbers that are expressed in decimal notation will either…
dvanaria
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