Assumptions Math Example 2

Solution

1. 1
   Assumption 1 (for contradiction): $\sqrt{2}$ is rational. This is the opposite of what we want to prove; the contradiction will disprove it.
2. 2
   Assumption 2: $p$ and $q$ are integers. This is what 'rational' means — expressible as a ratio of integers.
3. 3
   Assumption 3: $\gcd(p,q)=1$ (in lowest terms). This ensures the fraction is fully reduced; any rational number can be put in this form, so it is without loss of generality.
4. 4
   All three are necessary: without them the argument does not have enough structure to reach a contradiction.