Squares and Square Roots Notes

The square of a number n is n × n, written n². The numbers 1, 4, 9, 16, 25, … are perfect squares.

Useful properties: a perfect square never ends in 2, 3, 7, or 8. A perfect square ends in an even number of zeros. Squares of odd numbers are odd; squares of even numbers are even. The square of a number ending in 5 ends in 25.

The number of non-perfect-square numbers between n² and (n + 1)² is 2n. For example, between 9 (= 3²) and 16 (= 4²) there are 2 × 3 = 6 numbers.

The sum of the first n odd natural numbers is n². So 1 + 3 + 5 + 7 = 16 = 4².

For a number ending in 5, say a5: its square is a(a + 1) followed by 25. For example 35² = (3 × 4) | 25 = 1225, and 85² = (8 × 9) | 25 = 7225.

The identity (a + b)² = a² + 2ab + b² lets you square numbers close to a round number: 103² = 100² + 2(100)(3) + 3² = 10000 + 600 + 9 = 10609.

Three natural numbers a, b, c with a² + b² = c² form a Pythagorean triplet — the sides of a right-angled triangle. Examples: (3, 4, 5), (5, 12, 13), (8, 15, 17).

For any natural number m > 1, the triplet (2m, m² − 1, m² + 1) is Pythagorean. Taking m = 4 gives (8, 15, 17).

Multiplying a triplet by a constant gives another triplet, e.g. (6, 8, 10) from (3, 4, 5).

By prime factorisation: write the number as a product of primes, pair equal primes, and take one prime from each pair. For 324 = 2 × 2 × 3 × 3 × 3 × 3, √324 = 2 × 3 × 3 = 18.

By long division: this method also works for numbers that are not perfect squares and for decimals, giving the square root to any required number of places.

To make a number a perfect square, divide or multiply by the prime that is unpaired. For example, 90 = 2 × 3 × 3 × 5; multiplying by 2 × 5 = 10 gives 900 = 30².