Real Numbers — Formula Sheet

Examples of irrationals

√2, √3, √5, √7, π, 0.10110111011110...

The proof that √2 is irrational uses a method called 'Proof by Contradiction'.

STEP 1 — Assume the Opposite

Assume that √2 is rational.

STEP 2 — Write in Fraction Form

Since √2 is rational, we can write √2 = a/b, where a and b are integers, b ≠ 0, and HCF(a, b) = 1 (i.e., a and b are coprime).

STEP 3 — Square Both Sides

Squaring gives 2 = a²/b², so a² = 2b².

For example, 32760 = 2³ × 3² × 5 × 7 × 13.

Examples include √2, √3, √5, π, and 0.101101110...