# Square number

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 3^{2} and can be written as 3 × 3.

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The usual notation for the square of a number n is not the product *n* × *n*, but the equivalent exponentiation *n*^{2}, 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 *n*^{2}. 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).

Square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, 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 *n*^{2}, with 0^{2} = 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,
.

Starting with 1, there are square numbers up to and including m, where the expression represents the floor of the number x.