Irrational Number/Examples/Cube Root of 2

Example of Irrational Number
$\sqrt [3] 2$ is irrational.

Proof
$\sqrt [3] 2 = \dfrac m n$ for integers $m$ and $n$ such that:
 * $m \perp n$

where $\perp$ denotes coprimality.

Then:
 * $m^3 = 2 n^2$

Thus $2 \divides m^3$ and so $2 \divides m$.

Hence:
 * $m = 2 k$

for some $k \in \Z$.

Then:
 * $8 k^2 = 2 n^2$

and so $2 \divides n$.

But then we have $2 \divides m$ and $2 \divides n$

Hence $m$ and $n$ are not coprime after all.

From this contradiction the result follows.