Square numbers

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as $3 \times 3$ .

The usual notation for the square of a number $n$ is not the product $n \times n$ , but the equivalent exponentiation $n 2$ , usually pronounced as " $n$ squared". The name square number comes from the name of the shape; see below.

Square numbers are non-negative. Another way of saying that a (non-negative) integer is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number.

A positive integer that has no perfect square divisors except 1 is called square-free.

For a non-negative integer $n$ , the $n$ th square number is $n 2$ , with $02 = 0$ being the zeroth one. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, e.g., $\textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}$ .

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