A Square and A Cube
Square numbers
See why numbers like 1, 4, 9 and 16 are called square numbers.
Why are 1, 4, 9, 16, … called squares at all? Because of a shape.
The number of unit squares inside a square (its area) is its side times its side:
| Side | Unit squares |
|---|---|
| 1 | 1 |
| 2 | 2 × 2 = 4 |
| 3 | 3 × 3 = 9 |
| 4 | 16 |
| 5 | 25 |
| 10 | 100 |
So the number and the shape share a name. If you can lay out that many tiles as an actual square, the number is a square.
We write it short: 1 × 1 = 1² = 1, 2 × 2 = 2² = 4, 3 × 3 = 3² = 9, 4 × 4 = 4² = 16, 5 × 5 = 5² = 25.
In general, for any number n, n × n = n², read as 'n squared'.
A square's side doesn't have to be a whole number. Can a square have side 3/5 units, or 2.5 units? Yes. Their areas are (3/5)² = 3/5 × 3/5 = 9/25, and (2.5)² = 2.5 × 2.5 = 6.25.
The squares of the natural numbers (1, 2, 3, …) get a special name: perfect squares. So 1, 4, 9, 16, 25, … are perfect squares, while 6.25 is the square of something but not a perfect square. The lockers that stayed open were exactly the perfect-square ones.
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