The number 0.120120012000120000... (the gaps of zeros keep increasing) is:

**Useful results**:

1. The sum or difference of a rational and an irrational is IRRATIONAL.
Proof: if (rational + irrational) = rational, then irrational = rational − rational = rational. Contradiction.

2. The product of a NON-ZERO rational and an irrational is IRRATIONAL.

3. The sum/product of two irrationals can be RATIONAL or IRRATIONAL.
Example: $\sqrt{2}+\sqrt{2}=2\sqrt{2}$ (irrational); $\sqrt{2}+(-\sqrt{2})=0$ (rational).
Example: $\sqrt{2}\times \sqrt{2}=2$ (rational); $\sqrt{2}\times \sqrt{3}=\sqrt{6}$ (irrational).

These are the rules you cite when proving expressions like '$3+2\sqrt{5}$ is irrational'.