Number System — Quick Revision
Natural numbers (1, 2, 3, …) and whole numbers (0, 1, 2, …) are the starting point; integers include negatives. A rational number can be written as p/q where p and q are integers and q ≠ 0 — its decimal expansion is terminating or non-terminating recurring. An irrational number cannot be written as p/q; its decimal expansion is non-terminating and non-recurring, for example √2 and π. Together, rational and irrational numbers form the real numbers, which fill the entire number line. We can rationalise a denominator by multiplying by a suitable factor, and apply the laws of exponents to real numbers.
Key Points
- •Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real
- •Rational = p/q, q ≠ 0 (terminating or recurring decimals)
- •Irrational = non-terminating, non-recurring decimals (√2, π)
- •Every real number is a point on the number line
- •Rationalise by multiplying with the conjugate
- •aᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ