 # Confirm that if x is rational and x neq 0, then frac{1}{x} is rational. iohanetc 2021-02-13 Answered
Confirm that if x is rational and $x\ne 0$, then $\frac{1}{x}$ is rational.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it insonsipthinye
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.