Frank Day

2022-07-03

If $r\ne 0$ is rational and $i$ is irrational, then $ri$ is irrational?

amanhantmk

Expert

We only need to know that the rationals form a field: Suppose that $ri$ is rational (i.e. $ri\in \mathbb{Q}$)then $i=\frac{ri}{r}\in \mathbb{Q}$ because we can divide a rational by a non-zero rational and the result is rational. But $i$ is given not to be rational. This contradiction shows $ri$ is not rational, so irrational.