The product of a non-zero rational number and an irrational number is always:

**Sum/difference of two irrationals can be rational.** Example: $(1+\sqrt{2})+(1-\sqrt{2})=2$.
**Product of two irrationals can be rational.** Example: $\sqrt{2}\times \sqrt{2}=2$.
**Sum of a rational and an irrational is always irrational.** $3+\sqrt{2}$ is irrational.
**Product of a non-zero rational and an irrational is always irrational.** $5\times \sqrt{2}$ is irrational.

These facts let you build new irrationals and prove things irrational by contradiction.