Why "square"?
A square number is what you get when you multiply a whole number by itself. It’s called square because that many dots can be arranged in a perfect square.
9 dots → 3 rows of 3 → a 3-by-3 square.
The area of a square with side 3 cm is 3 × 3 = 9 cm². That’s where the word squared comes from.
Notation
7 × 7 is written 7², read "seven squared". The small 2 is an exponent: it tells you to multiply 7 by itself.
The first 15 perfect squares
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 9 | 81 |
| 2 | 4 | 10 | 100 |
| 3 | 9 | 11 | 121 |
| 4 | 16 | 12 | 144 |
| 5 | 25 | 13 | 169 |
| 6 | 36 | 14 | 196 |
| 7 | 49 | 15 | 225 |
| 8 | 64 | 20 | 400 |
A perfect square is a whole number whose square root is also a whole number. 49 is a perfect square because √49 = 7. 50 isn’t.
Patterns in square numbers
- Odd number steps: The differences between square numbers are the odd numbers: 1, 4, 9, 16, 25 go up by 3, 5, 7, 9.
- Last digits: Perfect squares only end in 0, 1, 4, 5, 6 or 9. A number ending in 2, 3, 7 or 8 can’t be a perfect square.
- Factors: Perfect squares have an odd number of factors (36 has 1, 2, 3, 4, 6, 9, 12, 18, 36).
- Prime factorization: every exponent is even: 36 = 2² × 3².
Squaring larger numbers
- 30² = 900 (3² = 9, then two zeros)
- 25² = 625. Trick for numbers ending in 5: multiply the tens digit by the next number up (2 × 3 = 6), then put 25 on the end → 625.
- 0.5² = 0.25
Squaring negative numbers
(−4)² = (−4) × (−4) = 16. A square is never negative. But −4² means −(4²) = −16, so brackets matter.
Worked examples
Example 1: What is 7²?
7 × 7 = 49.
Example 2: Is 144 a perfect square?
12 × 12 = 144. Yes.
Example 3: A square garden has area 81 m². How long is each side?
√81 = 9 m.
Common mistakes
- Doubling instead of squaring: 7² is 49, not 14.
- Forgetting brackets with negatives.
- Thinking all even numbers are perfect squares.
Try it yourself
- 9²
- 15²
- Is 200 a perfect square?
- Could 1,237 be a perfect square? Why?
- What is (−6)²?
Answers: 1) 81 2) 225 3) no 4) No: it ends in 7. 5) 36
Quick recap
- A square number is n × n, written n².
- Perfect squares have whole-number square roots.
- Squares are never negative and end only in 0, 1, 4, 5, 6 or 9.