iohanetc

## Answered question

2021-02-13

Confirm that if x is rational and $x\ne 0$, then $\frac{1}{x}$ is rational.

### Answer & Explanation

insonsipthinye

Skilled2021-02-14Added 83 answers

Given:
x is rational
$x\ne 0$
TO PROOF:$\frac{1}{x}$ is rational
DIRECT PROOF
Property rational numbers:
if a is a rational number, then there exists two integers y and z such that
$a=\frac{y}{z}\left(withz\ne 0\right)$.
x is an rational number and thus there exists integers y and z such that:
$x=\frac{y}{z}$
Since $x-0$, we then also know that the numerator cannot be zero
$y\ne 0$
We are interesred in $\frac{1}{x}$:
$\frac{1}{x}=\frac{1}{\frac{y}{z}}=\frac{z}{/}y$
Since z and y are integers(with $y\ne 0$), we then know that $\frac{1}{x}$ is rational.
Result: if x is rational and $x\ne 0$, then $\frac{1}{x}$ is rational.

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