# 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.

208 questions
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### Produce an explicit bijection between rationals and naturals?

I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but ...
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### How can I prove that all rational numbers are either terminating decimal or repeating decimal numerals?

I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly ...
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I stumbled upon this "relation" (is the name correct?): $$\lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&x\text{ is rational}\\ 0,&x\text{ is irrational}\end{... 9answers 23k views ### Proof that every repeating decimal is rational Wikipedia claims that every repeating decimal represents a rational number. According to the following definition, how can we prove that fact? Definition: A number is rational if it can be written ... 11answers 4k views ### What are the Laws of Rational Exponents? On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ... 1answer 2k views ### System of linear equations having a real solution has also a rational solution. I saw this question Let A ∈ M_{m\times n}(\mathbb{Q}) and b ∈ \mathbb{Q}^m. Suppose that the system of linear equations Ax = b has a solution in \mathbb{R}^n. Does it necessarily have a ... 3answers 6k views ### GCD of rationals Disclaimer: I'm an engineer, not a mathematician Somebody claimed that \gcd only is applicable for integers, but it seems I'm perfectly able to apply it to rationals also:$$ \gcd\left(\frac{13}{...
Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...