Perfect Squares – Numbers with Whole Square Roots

A perfect square is a whole number whose square root is also a whole number – 1, 4, 9, 16, 25, and so on, exactly the square numbers covered earlier in this section, now viewed from the opposite direction. Recognising perfect squares on sight is one of the most useful shortcuts in this whole section: it lets you instantly spot which square roots simplify to exact whole numbers, and which ones (covered on the Estimating Square Roots page next) need to be estimated instead.

A note on names: “perfect square” has nothing to do with the unrelated idea of a “perfect number” from number theory (a number like 6 or 28 whose divisors add up to itself) – the shared word “perfect” is a historical coincidence between two completely separate mathematical ideas.

Checking for a Perfect Square

Finding the Next Perfect Square

Real-Life Application

• Mental maths: quickly checking whether a square root will be a whole number.
• Simplifying radicals: spotting perfect square factors, covered later in this section.
• Geometry: designing a square layout with a whole-number side length from a given area.

Key Takeaways

Practice: Perfect Squares