Square number — Safekipedia Adventurer

In mathematics, a square number or perfect square is an integer that is the square of an integer. This means it is the product of a number multiplied by itself. For example, 9 is a square number because it equals 3², or 3 × 3.

We write the square of a number n as n², which we say as "n squared". The name comes from the shape of a square. The unit of area is the area of a unit square, which is 1 × 1. So, a square with side length n has area n².

In the real number system, square numbers are always zero or positive. An integer is a square number when its square root is also an integer. For example, 9 is a square number because its square root is 3, which is an integer.

Examples

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

The squares of the 21 integers centered at 0

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. 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

The sum of the first n odd integers is n2. 1 + 3 + 5 + ... + (2n − 1) = n2. Animated 3D visualization on a tetrahedron.

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

One key feature of square numbers is that they can be shown as points arranged in a square shape. Also, if you add up the first n odd numbers, you always get a square number. For instance, 1 + 3 + 5 + 7 + 9 = 25, which is 5². There are special patterns and links between square numbers and other types of numbers.

m = 1² = 1
m = 2² = 4
m = 3² = 9
m = 4² = 16
m = 5² = 25

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.

Special cases

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

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