The World of Numbers

Real numbers are rational (p/q) or irrational. Decimals that end or repeat are rational; surds like √2 are not. Exponent laws work for fractional powers too.

Key ideas

A rational number can be written p/q with integers p, q and q ≠ 0. Its decimal either ends or repeats.
In lowest terms, p/q gives an ending decimal only when q has no prime factors other than 2 and 5.
An irrational number has a decimal that never ends and never repeats, e.g. √2, √3, π.
Every real number is a point on the number line, and every point is a real number.
To rationalise 1/(√a + √b), multiply the top and bottom by √a − √b.
Fractional powers are roots: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ.

Remember

Exponent laws	aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ
Zero and negative powers	a⁰ = 1; a⁻ⁿ = 1/aⁿ
Surds	√a × √b = √(ab); (√a + √b)(√a − √b) = a − b
Repeating decimal	x = 0.272727…: 100x − x = 27, so x = 27/99 = 3/11

Watch out

Writing √(a + b) = √a + √b: √(9 + 16) = 5, not 7.
Forgetting to put the fraction in lowest terms before checking for 2 and 5.
Multiplying by √a + √b instead of the conjugate √a − √b.

Worked example

Write 73/80 in decimal form.

Working: 80 has only 2 and 5 as prime factors, so it terminates: 73 ÷ 80 = 0.9125

Answer: 0.9125