Real Numbers — Formula Sheet

Commutative Property (Multiplication)

a × b = b × a for all rational a, b.

Terminating decimals

denominator's prime factors are only 2 or 5 (e.g., 2/5 = 0.4, 101/8 = 12.625).

Non-terminating recurring

denominator has prime factors other than 2 or 5 (e.g., 17/36 = 0.4722…).

Examples

√2 = 1.41421…, √3 = 1.73205…, π = 3.14159…

Proof that √2 is irrational uses contradiction method (indirect proof) — very important for exams.

Examples include √2, √3, π, and (2 + √5).

We prove √2 is irrational using contradiction

Assume √2 = p/q (in reduced form).

Then 2q² = p², meaning p² is even, so p is even.

For example, √7, ∛5, and ⁴√8 are surds because their values are irrational.

For example

√48 = √(16 × 3) = √16 × √3 = 4√3.

Similarly, √98 = √(49 × 2)