MP Board · Class 10 · Mathematics · Real NumbersExplain why the number √2 cannot be expressed as a rational number.

The number √2 cannot be expressed as a rational number because it is irrational. If we assume it is rational, we can write it as a fraction a/b in simplest form where a and b are integers with no common factors other than 1, and b ≠ 0. Squaring both sides gives 2 = a²/b², meaning a² = 2b². This implies that 2 divides a², so 2 must also divide a. Letting a = 2c, we substitute it to get 4c² = 2b², which means b² = 2c². This shows 2 divides b as well, contradicting the assumption that a and b have no common factors.