Decomposition Math Example 4

Solution

1. 1
   Sub-goal 1: Assume $\sqrt{2}+\sqrt{3} = r$ is rational. Isolate: $\sqrt{3} = r - \sqrt{2}$.
2. 2
   Sub-goal 2: Square both sides: $3 = r^2 - 2r\sqrt{2} + 2$, so $2r\sqrt{2} = r^2-1$, giving $\sqrt{2} = \frac{r^2-1}{2r}$.
3. 3
   Sub-goal 3: If $r$ is rational, the right side is rational, so $\sqrt{2}$ would be rational — contradiction.
4. 4
   Conclusion: The original assumption is false; $\sqrt{2}+\sqrt{3}$ is irrational.