Square numbers
When a number is multiplied by itself, the result is its square: 5 × 5 = 25, so 25 is the square of 5, written 5².
A natural number that is the square of another natural number is a perfect square (or square number): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
Properties of perfect squares
These help you rule numbers out quickly:
A perfect square never ends in 2, 3, 7 or 8. So 1478 is not a perfect square.
If a number ends in 0, its square ends in 00. A perfect square always has an even number of zeros at the end, so 4000 is not a perfect square.
The square of an even number is even, and the square of an odd number is odd.
The last digit of a square depends only on the last digit of the number:
Number ends in | Square ends in |
|---|---|
1 or 9 | 1 |
2 or 8 | 4 |
3 or 7 | 9 |
4 or 6 | 6 |
5 | 5 |
0 | 0 |
Useful patterns
Sum of odd numbers: 1 + 3 + 5 + … + (first n odd numbers) = n². For example, 1 + 3 + 5 + 7 = 16 = 4².
Between consecutive squares: there are exactly 2n non-square numbers between n² and (n + 1)². Between 4² = 16 and 5² = 25 there are 2 × 4 = 8 numbers.
Squaring numbers ending in 5: for a number "a5", multiply a by (a + 1) and write 25 after it. 35² → 3 × 4 = 12, so 35² = 1225.
Pythagorean triplets
Three natural numbers a, b, c with a² + b² = c² form a Pythagorean triplet: for example 3, 4, 5 and 6, 8, 10.
For any natural number m > 1, the numbers 2m, m² − 1 and m² + 1 form a Pythagorean triplet.
Example. For m = 4: 8, 15 and 17. Check: 64 + 225 = 289 = 17². ✓
Square roots
Finding the square root is the inverse of squaring. Since 9² = 81, the square root of 81 is 9, written √81 = 9. (−9 also squares to 81, but in this chapter we take only the positive square root.)
Method 1: Repeated subtraction of odd numbers
Subtract 1, 3, 5, 7, … in order until you reach 0. The number of steps is the square root.
36 − 1 = 35, − 3 = 32, − 5 = 27, − 7 = 20, − 9 = 11, − 11 = 0. That is 6 steps, so √36 = 6.
Method 2: Prime factorisation
Write the number as a product of primes, pair up identical primes, and take one prime from each pair.
Example. √324.
324 = 2 × 2 × 3 × 3 × 3 × 3 = (2 × 2) × (3 × 3) × (3 × 3). Take one from each pair: 2 × 3 × 3 = 18.
If some prime is left without a partner, the number is not a perfect square.
Example. What is the smallest number that 1008 must be multiplied by to make it a perfect square?
1008 = (2 × 2) × (2 × 2) × (3 × 3) × 7. The 7 has no partner, so multiply by 7. Then 1008 × 7 = 7056 = 84².
Method 3: Estimation
For a number that is not a perfect square, find the two consecutive perfect squares around it.
Example. Estimate √250. Since 15² = 225 and 16² = 256, √250 lies between 15 and 16. It is much closer to 256, so √250 ≈ 15.8.
Quick summary
Perfect squares never end in 2, 3, 7 or 8, and they have an even number of trailing zeros.
The sum of the first n odd numbers is n².
2m, m² − 1 and m² + 1 form a Pythagorean triplet.
Square root by prime factorisation: pair the primes and take one from each pair.