Square number

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 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.

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, 9 is a square number because its square root is 3, an integer.

Examples

Here are some square numbers smaller than 60² which equals 3600:

0² = 0
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100

Square numbers are special because they come from multiplying a number by itself. For example, 9 is a square number because it is 3 × 3. You can find more square numbers by continuing this pattern.

Properties

A square number is an integer that is the square of another integer. This means it can be written as some number multiplied by itself. For example, 9 is a square number because it equals 3 × 3, or 3².

One key property of square numbers is that they can be shown as points arranged in a square shape. Also, the sum of the first n odd numbers always gives a square number. For instance, 1 + 3 + 5 + 7 + 9 = 25, which is 5². There are also special patterns and relationships between square numbers and other types of numbers.

Odd and even square numbers

Squares of even numbers are even and can be divided by 4. For example, 4 (which is 2×2) and 16 (which is 4×4) are both even and divisible by 4.

Squares of odd numbers are odd. When you divide an odd square number by 8, the remainder is always 1. For example, 9 (which is 3×3) divided by 8 leaves a remainder of 1. Every odd square number is also a centered octagonal number. The difference between any two odd square numbers is always a multiple of 8.

Special cases

Square numbers have some interesting patterns. For example, if a number ends in 5, its square will also end in 25. If a number ends in 6, its square will end in 36. These special ending numbers are called automorphic numbers and follow predictable patterns.

There are also quick ways to figure out the square of certain numbers. For numbers that end in 0, their squares simply end in two zeros. For numbers that end in 5, their squares always end in 25. These patterns help make calculating squares easier and faster!