Square number information

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 $3 \times 3$ .

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. The unit of area is defined as the area of a unit square ( $1 \times 1$ ). 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).

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, ${\sqrt {9}}=3,$ 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 $n$ 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, for example, $\textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}$ .

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

^ Some authors also call squares of rational numbers perfect squares.

[1] Some authors also call squares of rational numbers perfect squares.

Square number information

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Squaring the square