02 / 10 · Square roots
Perfect squares
Why some roots are whole numbers
Some numbers of tiles can be arranged into a perfect square: 1, 4, 9, 16, 25… These are the perfect squares, and their roots are whole numbers.
Most numbers are not. Twenty tiles make a 4 × 4 square with 4 tiles left over. That tells you straight away that √20 is more than 4 but less than 5.
20 = 4² + 4 → 4 < √20 < 5 25 = 5² + 0 → √25 = 5 (n + 1)² − n² = 2n + 1
Try it · Change the number of tiles. Orange tiles form the biggest square possible; grey tiles are left over; outlines show what is missing for the next square.
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There is a pattern in how squares grow. To go from a 4 × 4 square to a 5 × 5 one, you add a row, a column and one corner tile: 4 + 4 + 1 = 9 tiles. Going from n × n to (n + 1) × (n + 1) always takes 2n + 1 tiles, which is why the gaps between perfect squares keep getting bigger: 1, 3, 5, 7, 9…
Knowing the perfect squares by heart is the first step to estimating any root in your head.
Key ideas
Words worth remembering
- Perfect square
- A whole number that is some whole number squared, like 36 = 6².
- Bracketing
- Trapping a root between two whole numbers using the nearest perfect squares.
Think about it
How many tiles would you need to add to 50 to reach the next perfect square?