Square number

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In mathematics, a square number or perfect square is an integer that is the square of an integer;^[1] in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 3² and can be written as 3 × 3.

The usual notation for the square of a number n is not the product n × n, but the equivalent exponentiation n², usually pronounced as "n squared". The name square number comes from the name of the shape. The unit of area is defined as the area of a unit square (1 × 1). Hence, a square with side length n has area n². If a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers).

In the real number system, square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, ${\displaystyle {\sqrt {9}}=3,}$ so 9 is a square number.

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

For a non-negative integer n, the nth square number is n², 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, for example, ${\displaystyle \textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}}$.

Starting with 1, there are ${\displaystyle \lfloor {\sqrt {m}}\rfloor }$ square numbers up to and including m, where the expression ${\displaystyle \lfloor x\rfloor }$ represents the floor of the number x.