Square number facts for kids

A square number, sometimes also called a perfect square, is the result of an integer multiplied by itself. 1, 4, 9, 16 and 25 are the first five square numbers. In a formula, the square of a number n is denoted n² (exponentiation), usually pronounced as "n squared". The name square number comes from the name of the shape; see below.

Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number.

Examples

The squares (sequence A000290 in OEIS) smaller than 70² are:

There are infinitely many square numbers, as there are infinitely many natural numbers.

Properties

The number m is a square number if and only if one can compose a square of m equal (lesser) squares:

m = 12 = 1
m = 22 = 4
m = 32 = 9
m = 42 = 16
m = 52 = 25
Note: White gaps between squares serve only to improve visual perception. There must be no gaps between actual squares.

A square with side length n has area n².

The expression for the nth square number is n². This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows:

So for example, 5² =25= 1 + 3 + 5 + 7 + 9.

A square number can end only with digits 0, 1, 4, 6, 9, or 25 in base 10, as follows:

1. If the last digit of a number is 0, its square ends in an even number of 0s (so at least 00) and the digits preceding the ending 0s must also form a square.
2. If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four.
3. If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
4. If the last digit of a number is 3 or 7, its square ends in 9 and the number formed by its preceding digits must be divisible by four.
5. If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
6. If the last digit of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56.

A square number cannot be a perfect number.

All fourth powers, sixth powers, eighth powers and so on are perfect squares.

Special cases

• If the number is of the form m5 where m represents the preceding digits, its square is n25 where n = m × (m + 1) and represents digits before 25. For example the square of 65 can be calculated by n = 6 × (6 + 1) = 42 which makes the square equal to 4225.
• If the number is of the form m0 where m represents the preceding digits, its square is n00 where n = m². For example the square of 70 is 4900.
• If the number has two digits and is of the form 5m where m represents the units digit, its square is AABB where AA = 25 + m and BB = m². Example: To calculate the square of 57, 25 + 7 = 32 and 7² = 49, which means 57² = 3249.

Odd and even square numbers

Squares of even numbers are even (and in fact divisible by 4), since (2n)² = 4n².

Squares of odd numbers are odd, since (2n + 1)² = 4(n² + n) + 1.

It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.

As all even square numbers are divisible by 4, the even numbers of the form 4n + 2 are not square numbers.

As all odd square numbers are of the form 4n + 1, the odd numbers of the form 4n + 3 are not square numbers.

Squares of odd numbers are of the form 8n + 1, since (2n + 1)² = 4n(n + 1) + 1 and n(n + 1) is an even number.

• Eric W. Weisstein, Square Number at MathWorld.

• Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30–32, 1996. ISBN: 0-387-97993-X

• Dario Alpern, Sum of squares. A Java applet to decompose a natural number into a sum of up to four squares.

Images for kids

• 

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

See also

In Spanish: Cuadrado perfecto para niños