Square numbers
Theory
A square number is a whole number multiplied by itself, like \(5^2 = 5 \times 5 = 25\). That many dots can be arranged in a filled square. The first square numbers are \(1, 4, 9, 16, 25, 36, \ldots\)
To square a number, multiply it by itself. We write \(5^2 = 5 \times 5 = 25\); the small \(2\) means "two of them multiplied".
A square number is the answer: a whole number times itself. That many dots make a square array of \(n\) rows with \(n\) dots in each row.
The first square numbers are \(1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100,\ \ldots\)
The gaps between consecutive square numbers are the odd numbers \(3,\ 5,\ 7,\ 9,\ \ldots\) Each new square adds a row and a column of dots.
List a whole number, multiply it by itself, and read off the square number.
| Number \(n\) | \(n \times n\) | Square \(n^2\) |
|---|---|---|
| \(1\) | \(1 \times 1\) | \(1\) |
| \(2\) | \(2 \times 2\) | \(4\) |
| \(3\) | \(3 \times 3\) | \(9\) |
| \(4\) | \(4 \times 4\) | \(16\) |
| \(5\) | \(5 \times 5\) | \(25\) |
| \(6\) | \(6 \times 6\) | \(36\) |
| \(10\) | \(10 \times 10\) | \(100\) |
How to work with square numbers
- To square a number, multiply it by itself: \(n^2 = n \times n\).
- To check if a number is square, look for a whole number that gives it when multiplied by itself. If none does, it is not square.
- To continue the pattern, square the next whole number in turn: \(1^2, 2^2, 3^2, \ldots\)
- To find a gap, subtract one square from the next. The gaps are the odd numbers \(3, 5, 7, 9, \ldots\)
Multiply \(9\) by itself, so the square of \(9\) is \(81\).
| \(9^2 = 9 \times 9\) | \(=\) | \(81\) |
Test whole numbers times themselves. Only \(49\) works, so the answer is \(49\).
| \(7 \times 7\) | \(=\) | \(49\) |
These are \(3^2, 4^2, 5^2\), so the next is \(6^2\). The next square number is \(36\).
| \(6 \times 6\) | \(=\) | \(36\) |
Subtract the smaller square from the larger. The gap is the odd number \(15\).
| \(64 - 49\) | \(=\) | \(15\) |
Common pitfalls
Frequently asked questions
What is a square number?
A whole number multiplied by itself. \(1, 4, 9, 16, 25\) and \(36\) are square numbers because \(1\times1, 2\times2, 3\times3\) and so on give them.
What does squaring a number mean?
Multiplying it by itself. \(7\) squared is written \(7^2\) and equals \(7 \times 7 = 49\).
How can I tell if a number is a square number?
Look for a whole number that gives it when multiplied by itself. \(64\) is square because \(8 \times 8 = 64\); \(45\) is not, because it sits between \(36\) and \(49\).
Why is it called a square number?
Because that many dots or tiles can be arranged in a filled square, with the same number of rows as columns. \(36\) makes a \(6 \times 6\) square.
Why do the gaps between squares grow by odd numbers?
Each new square adds a row and a column of dots that share one corner, so the extra dots are the next odd number. The gaps run \(3, 5, 7, 9, \ldots\)