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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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Two queries on triangles whose side lengths are rational

Let us define a 'rational triangle' as one with lengths of all sides rational. We are aware that a positive integer is called 'congruent' only if it is the area of a RIGHT triangle with rational length sides; so we have, every rational number is…
Nandakumar R
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Which of the statements are false?

I have this statement: Let $a, b, c, d \in \mathbb{R} - $ {$0$}, with $\quad acd> 0$. If $– 1 < \frac{a}{b} < \frac{b}{c} < \frac{c}{d} < \frac{d}{a} < 1,$ Which of the following alternative are false? A) $\quad a < cd$ B) $\quad c < ad$ C) $\quad…
ESCM
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Means and Set of Rational Numbers

Let $S$ be the set $\{0, 1\}$. Given any subset of $S$ we may add its arithmetic mean to $S$ (provided it is not already included - $S$ never includes duplicates). Show that by repeating this process we can include the number $1/5$ in $S$. Show that…
Saurabh Shukla
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What numbers can be approximated by ratios of numbers containing only specified digits?

Let $D$ be a subset of the decimal digits $ \{ 0, 1, 2, \ldots, 9\}$, with $D \neq \{0\}$ or $\emptyset$. Let $N$ be the set of positive integers whose decimal representations (without leading $0$'s) consist only of digits in $D$. Let $R$ be the…
Dale
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Conditions for simplified rational expressions

$$\frac{x^2+6x+5}{x^2-x-2}$$ $$\frac{(x+5)(x+1)}{(x-2)(x+1)}$$ $$\frac{x+5}{x-2}$, $ x \ne -1$$ My question is when it comes to specifying that $ x \ne -1$, the end result is also undefined where $x=2$, but I only need to state the condition for…
altec
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Two reals with rational quotient, Z-span the integers

Given two real numbers $\alpha_1,\alpha_2\in\mathbb{R}\cap(0,1)$, with rational ratio $\frac{\alpha_1}{\alpha_2} \in \mathbb{Q}$, show there exist $m,n\in \mathbb{Z}$, such that $n\alpha_1 + m\alpha_2 \in \mathbb{Z}$, but $n\alpha_1 \notin…
Teddy
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At least one rational is within interval (A, B)

I'm reading a book and there is some strange proof (strange for me) of the theorem that within each interval, no matter how small, there are rational points. Proof: we need only take a denominator $n$ large enough so that the interval $[0,…
E. Shcherbo
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Quotient vs Ratio vs Fraction vs Rational

I can't seem to differentiate between a Quotient and a Ratio and a Fraction and a Rational. From what I know a rational is a number like $2/3$ or $5.4/7$ whereas quotient, ratio and fraction all are just three names for one same thing $1+x/1-x^3$ ,…
ArnuldOnData
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Is there any proper way through which we can check whether a number is rational or not

Is there any proper way through which we can check whether a number is rational or not because finding the decimal expansion at times is not viable. For example the number $48/121$ shows no sign of termination or repetition of its digits well to…
chemophilic
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Identifying if the rational number terminates or repeats by looking at numerator and denominator

I was wondering if a rational number p/q can be identified as either repeating decimal or terminating decimal by looking at numerator and denominator. In other words, is there a property that if p and q follow, then p/q is always repeating decimal.
Amardeep reddy
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Is $"2.1234........ "$ rational?

In my excercise book of math , I have found one problem . In that problem I have been asked to detect whether the number $2.1234....$ is rational or irrational? My concept is : "$2.1234....$ is irrational." But the answer of book says that the…
Way to infinity
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Rational numbers via equivalent classes

Sometimes the rational numbers $\mathbb{Q}$ are defined via equivalent classes $[(a.b)]\subset\mathbb{Z}\times\mathbb{Z}$ of integers. In general we have $(a_1,b_1)\sim (a_2,b_2):\Leftrightarrow a_1b_2=a_2b_1$. How does such a class $[(a,b)]$ looks…
user337073
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Multiplication and division of a pair of real numbers

Suppose that you have any two real numbers: $x_1$ and $x_2$ (different from $0$). If we know that the product $x1x2$ is rational and the division $\frac{x_1}{x_2}$ is also rational, is it possible to show that $x_1$ and $x_2$ are rational as…
Marcos
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cross multiplication property

good morning guys,i have such question,when i was reading GRE book,there was such kind of property related to rational number,in shortly if we are trying to determine if $a/b$ is more then $c/d$ or vice verse,there was explained very…
dato datuashvili
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Adding Rational Expressions

$$\frac{7a}{6a^2-15a} + \frac{12a}{4a^2-25}$$ I determined the LCD of the denominators: $(3a)(2a-5)(2a+5)$. I then multiplied all nominators by the LCM, combined the terms and got: $$\frac{a(86a+95)}{(3a)(2a-5)(2a+5)}$$ Where did I go wrong?
Grimestock
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