Square Numbers and Perfect Squares

A square number, also called a perfect square, is the result you get when you multiply a whole number by itself. If \( n \) is a whole number, then its square number is \( n \times n = n^2 \). For example, \( 5 \times 5 = 25 \), so 25 is a square number, and \( 5^2 \) is read as "5 squared."

Square numbers show up naturally inside a multiplication table. If you have already worked with multiplication tables, you have likely noticed that the diagonal of the table (\(1 \times 1\), \(2 \times 2\), \(3 \times 3\), and so on) is made up entirely of square numbers.

Here is a perfect squares chart showing the square of every whole number from 1 to 12. Memorizing this list makes it much faster to recognize perfect squares later on.

Square numbers get their name from geometry. If you arrange dots into rows and columns that have the same number of dots in each direction, you form an actual square shape. Below, 4 rows of 4 dots form a 4-by-4 square, which is exactly \( 4 \times 4 = 16 \) dots in total.

To find the square of any whole number, simply multiply that number by itself: \( n^2 = n \times n \). For small numbers, you can read the answer straight from a multiplication table. For larger numbers, it can help to use mental multiplication strategies, such as breaking a number into tens and ones before multiplying. For example, to find \( 13^2 \), think of 13 as \( 10 + 3 \), so \( 13^2 = 13 \times (10 + 3) = 130 + 39 = 169 \).

Square numbers follow a neat pattern: the difference between one square number and the next is always an odd number, and these odd numbers increase by 2 each time.

Look at the sequence \(1, 4, 9, 16, 25, 36\). The differences are \( 4 - 1 = 3 \), \( 9 - 4 = 5 \), \( 16 - 9 = 7 \), \( 25 - 16 = 9 \), and \( 36 - 25 = 11 \). Notice that the differences, \(3, 5, 7, 9, 11\), are exactly the odd numbers in order. This pattern is a quick way to check whether a chart of perfect squares has been built correctly.

Plotting \( n \) against \( n^2 \) shows how quickly square numbers grow compared to the numbers being squared. The curve starts out flat for small values of \( n \) and then rises steeply, because each square number grows by a larger amount than the one before it.

Example 1: Find the square of 7.

Using the definition of a square number, \( 7^2 = 7 \times 7 = 49 \). So 49 is a perfect square.

Example 2: Is 50 a perfect square?

Check the perfect squares chart: \( 7^2 = 49 \) and \( 8^2 = 64 \). Since 50 falls between 49 and 64, and it does not match \( 7^2 \) or \( 8^2 \) exactly, 50 is not a perfect square.

Example 3: Find the next square number after \(9^2 = 81\).

Using the odd-number pattern, the next difference after \(11\) (from \(81 - 64\)) is \(13\), so the next square number is \(81 + 17\)... instead, simply calculate directly: \( 10^2 = 100 \). Checking with the pattern, \( 100 - 81 = 19 \), which is indeed the next odd number after \(17\), confirming the result.

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