We can prove this by "Proof by contradiction", and we can find a contradiction through arithmetic basis.
[tex]\text{Assume } \sqrt{5} - \sqrt{3} \text{ is rational.}[/tex]
[tex]\text{That is, } \sqrt{5} - \sqrt{3} = \frac{a}{b}\text{, where a and b do not have any common factors.}[/tex]
[tex]\text{Squaring both sides: } (\sqrt{5} - \sqrt{3})^{2} = \frac{a^{2}}{b^{2}}[/tex]
[tex]5 - 2\sqrt{15} + 3 = \frac{a^{2}}{b^{2}}[/tex]
[tex]8 - 2\sqrt{15} = \frac{a^{2}}{b^{2}}[/tex]
[tex]2(4 - \sqrt{15}) = \frac{a^{2}}{b^{2}}[/tex]
We can prove by contradiction based on the fact that the square root of 15 is irrational. We've made our assumption that we can write √5 - √3 in fraction form. By making √15 the subject, we would have contradicted our original assumption.
[tex]4 - \sqrt{15} = \frac{a^{2}}{2b^{2}}[/tex]
[tex]\sqrt{15} = 4 - \frac{a^{2}}{2b^{2}}[/tex]
Now we've hit our jackpot. Since we can write √15 in a rational form, we've contradicted ourselves. This implies that our original assumption was wrong, which was that we can write √5 - √3 in fraction form. This further implicates that √5 - √3 cannot be rewritten in simplified fraction form, which means that √5 - √3 is irrational.
Thus, our proof is complete.
