Square Numbers – Multiplying a Number by Itself

A square number is what you get when you multiply a whole number by itself: 1, 4, 9, 16, 25, and so on. The name comes from geometry – if you arrange that many dots into a perfect square grid, the side length is exactly the number you squared. 16 dots arrange into a neat 4-by-4 square; that's exactly why 4² = 16 is called a “square” number.

This geometric way of thinking about numbers goes back to the Pythagoreans of ancient Greece, around 500 BCE, who literally arranged pebbles into shapes to study numbers – a field they called “figurate numbers.” Square numbers were one of their favourite patterns, and the visual proof is still one of the best ways to understand square numbers today: you can literally see why they grow the way they do by watching the square grid expand outward, one new L-shaped layer of dots at a time.

The Pattern of Square Numbers

Squaring means multiplying a number by itself: n² = n × n. The first few square numbers are 1, 4, 9, 16, 25, 36, 49…

Find 13 squared.

13² = 13 × 13 = 169.

Recognising Square Numbers

Is 45 a square number?

The nearest square numbers are 36 (6²) and 49 (7²). 45 sits between them, so it is not a square number.

Real-Life Application

Area: the area of a square with side n is n².
Chessboards and grids: an 8×8 board has 8² = 64 squares.
Screen resolutions: some display formats use square-number pixel counts.

Key Takeaways

A square number is n × n for a whole number n.
The name comes from arranging that many objects into a square grid, an idea from the ancient Pythagoreans.
Square roots, covered later in this section, are the reverse operation.

Practice: Square Numbers

Find the Square Number