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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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Is there such an $x$ that both $2^{\frac{x}{3}}$ and $3^{\frac{x}{2}}$ are simultaneously rational?

I'm having difficulty with the following problem: Is there such an $x$ that both $2^{\frac{x}{3}}$ and $3^{\frac{x}{2}}$ are simultaneously rational, for $0
EngineerInProgress
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Infinite number of rationals between any two reals.

Let $a$ and $b$ be reals with $a
Jon
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High-school level proof that if $x_i$ are rationals and $\sqrt{x_1}+\dots\sqrt{x_n}$ is rational, then each $\sqrt{x_i}$ is rational.

I'm having a hard time with the following problem: Let $x_1,x_2...x_n$ be rational numbers. Prove that if the sum $\sqrt{x_1}+\sqrt{x_2}+...+\sqrt{x_n}$ is rational, then all $\sqrt{x_i}$ are rational. Show, that the assumption for $x_i$ to be…
EngineerInProgress
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Is there a function that gives the same result for a number and its reciprocal?

Is there a (non-piecewise, non-trivial) function where $f(x) = f(\frac{1}{x})$? Why? It would be nice to compare ratios without worrying about the ordering of numerator and denominator. For example, I might want to know whether the "magnitude" of…
Nolan Amy
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Prove that the square root of any irrational number is irrational.

The problem I'm having with this proof is that I'm not sure if my proof actually proves the theorem correct or if I'm using circular reasoning. Theorem: Prove that the square root of any irrational number is irrational. Proof: => Suppose not. The…
HLM
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Set theory and rational numbers

A set $Q$ contains $0$, $1$ and the average of all elements of every finite non-empty subset of $Q$. Prove that $Q$ contains all rational numbers in $[0,1]$. This is the exact wording, as it was given to me. Obviously, the elements that correspond…
Carlos Lopez
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Counting Real Numbers

Forgive me if this is a novice question. I'm not a mathematics student, but I'm interested in mathematical philosophy. Georg Cantor made an argument that the set of rational numbers is countable by showing a correspondence to the set of natural…
Haitham Gad
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Patterns in $\frac{80}{81}$ and $\frac{10}{81}$.

The decimal form of $\frac{80}{81}$ is $0.987654320\ldots$ notice the expected $1$ is missing. The decimal form of $\frac{10}{81}$ is $0.12345679\ldots$ notice the expected $8$ is missing. Can someone expansion why the decimal form is the way they…
Ahmed S. Attaalla
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Can rational numbers have decimals?

I had a question in my exam paper - Which of the following is not a rational number? a) $\sqrt{25}$ b) $\sqrt{45}$ c) $\sqrt\frac{256}{225}$ d) $\frac{3}{4}$ The answer to this is b. Now, $\sqrt{45} \approx 6.708$. Can someone explain why this not…
GreatBlitz
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Prove that $x\in\mathbb Q$

Let $a\in\mathbb Q$ and $a>\dfrac43$. Let $x\in\mathbb R$ and $x^2-ax,x^3-ax\in\mathbb Q$. Prove that $x\in\mathbb Q$. EDIT: Thsi is my attempt: Let $x^2-ax=b$ and $x^3-ax=q$ for some $b,q\in\mathbb{Q}$. Then I tried to write $x^2$ and $x^3$ in…
user164524
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HINT: Prove there does not exist a rational number that satisfies $x^3=p/q$

I would like to request a hint for a problem I am working on form Hardy's a Course of Pure Mathematics. Question Prove generally that a rational fraction $p/q$ in its lowest terms cannot be the cube of a rational function unless $p$ and $q$ are…
GovEcon
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Confused about incompleteness of rational numbers

The following two facts about rational numbers confuse me a bit: Rational numbers are ordered but unlike real numbers are not complete. The incompleteness is due to the existence of gaps in the rational numbers. When we say gaps for this ordered…
abk
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How to estimate the number of decimal places required for a division?

Given two decimal numbers, is it possible to estimate the number of decimal places required to fit the result of their division? Provided that the division yields a finite number of decimals, of course. For example: 1234.5678 / 2 = 617.2839, 4…
BenMorel
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If $x$ is a rational number, then $1/x$ is a rational number

Why is this statement false? If $x$ is a rational number, i.e. $\frac{p}{q}$, then shouldn't it be obvious that $\frac{q}{p}$ is also a rational number, by definition of rational numbers?
Jason
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$n ^ 2 +5 $, $n ^ 2 +10 $ are rational number square

Seeking a nonzero rational number $n$, such that $n ^ 2 +5 $, $n ^ 2 +10 $ are rational number square。 This is a high school students asked the question, answer $n=\frac{31}{12}$, but no answer process. I try to follow Parametrization of a…
geometryscience
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