Can we ever get an irrational number by dividing two rational numbers?

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If we try to divide any two random arbitrarily long rational numbers like

103850.2387209375029375092730958297836958623986868349693868398659825528365...

and

127.123123123...

Is it guaranteed that the result is also a rational number?

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2 Answers

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The quotient of two rationals is always a rational.

For if $\alpha = \frac{a}{b}$ and $\beta = \frac{c}{d}$ with $a, b, c, d$ integers with none of $b, c, d$ being zero, then

$$\frac{\alpha}{\beta} = \frac{ad}{bc}$$

is a quotient of integers, and so is rational.

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If we get an irrational number by dividing a rational number by another rational number, then product of rationals won't be a binary operation in $\mathbb{Q}$. But we know that $\mathbb{Q}\setminus \{0\}$ is a group with respect to multiplication.

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