Number Systems Notes

A rational number can be written as p/q where p and q are integers and q ≠ 0. Between any two rational numbers there are infinitely many rational numbers, so the rationals are 'dense'.

An irrational number cannot be written in the form p/q. Examples are √2, √3, √15, π, and 0.101001000100001… (a non-terminating, non-recurring decimal). The square root of any positive integer that is not a perfect square is irrational.

Rational and irrational numbers together make up the real numbers, and every real number corresponds to exactly one point on the number line, and vice versa.

The decimal expansion of a rational number is either terminating (e.g. 7/8 = 0.875) or non-terminating recurring (e.g. 1/3 = 0.333… = 0.3̄).

A rational number p/q in lowest terms has a terminating decimal expansion exactly when the prime factorisation of q contains only 2s and/or 5s. Otherwise the expansion is non-terminating recurring.

The decimal expansion of an irrational number is non-terminating and non-recurring. To convert a recurring decimal like x = 0.6̄ to a fraction: 10x = 6.6̄, so 10x − x = 6, giving x = 6/9 = 2/3.

The sum, difference, product, and quotient of two rational numbers is rational. But the sum or product of a rational and an irrational number is irrational (e.g. 2 + √3, 5√2). The sum of two irrationals may be rational (e.g. (2+√3) + (2−√3) = 4) or irrational.

To rationalise a denominator of the form 1/√a, multiply numerator and denominator by √a. To rationalise 1/(a + √b), multiply by the conjugate (a − √b), using (a + √b)(a − √b) = a² − b.

These identities are used constantly: (√a)² = a for a ≥ 0, √(ab) = √a·√b, and √a/√b = √(a/b) for a ≥ 0, b > 0.

For a positive real number a and rational numbers m and n, the exponent laws hold: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, and aᵐ · bᵐ = (ab)ᵐ.

Rational exponents extend roots: a^(1/n) means the positive nth root of a, and a^(m/n) = (a^(1/n))ᵐ = (aᵐ)^(1/n).

For example, 8^(2/3) = (8^(1/3))² = 2² = 4, and 5^(1/2) · 5^(1/2) = 5^(1) = 5.