A square number is the result of multiplying a whole number by itself. Its square root is the value that was multiplied. For example, 7 × 7 = 49, so 49 is a square number and the square root of 49 is 7.
What is a square number?
A square number (also called a perfect square) is any integer that can be written as another integer multiplied by itself.
Written in index notation: n² = n × n
The name comes from geometry — if you arrange n² dots into a grid, they form a perfect square shape with n rows and n columns.
The first 15 square numbers
| n | n² | Read as |
|---|---|---|
| 1 | 1 | "one squared" |
| 2 | 4 | "two squared" |
| 3 | 9 | "three squared" |
| 4 | 16 | "four squared" |
| 5 | 25 | "five squared" |
| 6 | 36 | "six squared" |
| 7 | 49 | "seven squared" |
| 8 | 64 | "eight squared" |
| 9 | 81 | "nine squared" |
| 10 | 100 | "ten squared" |
| 11 | 121 | "eleven squared" |
| 12 | 144 | "twelve squared" |
| 13 | 169 | "thirteen squared" |
| 14 | 196 | "fourteen squared" |
| 15 | 225 | "fifteen squared" |
In KS3 exams, you are normally expected to recall the squares from 1² to 15² without a calculator.
What is a square root?
The square root of a number is the value that, when multiplied by itself, gives that number. The symbol for square root is √.
√49 = 7 because 7 × 7 = 49
Every positive square number has two square roots — one positive and one negative — because (−7) × (−7) = 49 as well. At KS3, questions usually ask only for the positive square root.
Worked examples
Example 1 — find a square number
Work out 13².
13² = 13 × 13 = 169
Check: 13 × 13 = 13 × 10 + 13 × 3 = 130 + 39 = 169. ✓
Example 2 — find a square root of a perfect square
Find √144.
Ask: which number multiplied by itself gives 144?
12 × 12 = 144, so √144 = 12
Example 3 — decide whether a number is a square number
Is 81 a square number?
9 × 9 = 81, so yes, 81 is a square number (9²). ✓
Is 50 a square number?
7 × 7 = 49 and 8 × 8 = 64. There is no whole number between 7 and 8, so 50 is not a perfect square.
Example 4 — estimating a square root between two whole numbers
Estimate √70 to one decimal place.
8² = 64 and 9² = 81, so √70 lies between 8 and 9.
70 is 6 units above 64 (which spans 81 − 64 = 17 units total).
Rough estimate: 8 + 6/17 ≈ 8 + 0.35 ≈ 8.4
(Calculator check: √70 ≈ 8.366… so rounding to 8.4 is reasonable.)
Recognising square numbers in context
Square numbers appear across many KS3 topics:
- Pythagoras' theorem — you square the two shorter sides and add them: a² + b² = c²
- Area of a square — a square with side length n has area n² cm²
- Simplifying surds (GCSE bridge) — knowing perfect squares helps simplify expressions like √48 = √(16 × 3) = 4√3
Negative numbers and squares
A negative number squared is always positive:
(−5)² = (−5) × (−5) = 25
This is a common error source. Note that −5² ≠ (−5)² because the order of operations applies: −5² means −(5²) = −25.
What the national curriculum says
The DfE's KS3 mathematics programme of study requires pupils to use and understand integer powers and associated real roots, and to recognise and use square numbers. The AQA GCSE Mathematics specification confirms these are core recall topics for Year 7 and Year 8 and appear regularly in GCSE foundation and higher papers.
Frequently asked questions
What is the difference between a square number and a square root?
A square number is the result of multiplying an integer by itself — for example, 36 is a square number because 6 × 6 = 36. A square root is the reverse operation: the square root of 36 is 6 because 6² = 36. Squaring and taking a square root are inverse operations, just as multiplication and division are inverses.
Is 0 a square number?
Yes. 0 = 0 × 0 = 0², so 0 is technically a perfect square. However, KS3 exam questions almost always start the list of square numbers at 1. The square root of 0 is 0.
How do I check quickly whether a number could be a square number?
Square numbers always end in 0, 1, 4, 5, 6, or 9 — never in 2, 3, 7, or 8. This is a fast filter: if a number ends in 2, 3, 7, or 8, it cannot be a perfect square. If it passes that test, check by finding the square root and verifying whether it is a whole number.
Why does √49 have two answers (+7 and −7)?
Because both 7 × 7 = 49 and (−7) × (−7) = 49. The symbol √ by itself denotes the principal (positive) square root, so √49 = 7. In KS3 contexts you will write just the positive root unless a question specifically asks for both.
Do I need to memorise square numbers for KS3?
Yes. The DfE national curriculum expects you to know integer square numbers at KS3, and KS3 assessments regularly test recall of squares up to 15² (= 225) without a calculator. Knowing them also speeds up Pythagoras calculations and simplifying fractions with squared terms.
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