6'' Squares

In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. more...

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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.

A positive integer that has no perfect square divisors except 1 is called square-free.

The usual notation for the formula for the square of a number n is not the product n Ã— n, but the equivalent exponentiation n², usually pronounced as "n squared". For a non-negative integer n, the nth square number is n², with 0² = 0 being the zeroth square. 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 (e.g., 4/9 = (2/3)²).

Starting with 0, there are 1 + ⌊√m⌋ square numbers up to and including m.

Examples

The first 50 squares of natural numbers (sequence A000290 in OEIS) are:

Properties

The number m is a square number if and only if one can arrange m points in a square:

The formula 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 (marked as '+'). So for example, 5² = 25 = 1 + 3 + 5 + 7 + 9.

The nth square number can be calculated from the previous two by doubling the (n âˆ’ 1)-th square, subtracting the (n âˆ’ 2)-th square number, and adding 2, because n² = 2(n − 1)² − (n − 2)² + 2. For example, 2×5² âˆ’ 4² + 2 = 2×25 âˆ’ 16 + 2 = 50 âˆ’ 16 + 2 = 36 = 6².

It is often also useful to note that the square of any number can be represented as the sum 1 + 1 + 2 + 2 + ... + n âˆ’ 1 + n âˆ’ 1 + n. For instance, the square of 4 or 4² is equal to 1 + 1 + 2 + 2 + 3 + 3 + 4 = 16. This is the result of adding a column and row of thickness 1 to the square graph of three (like a tic tac toe board). You add three to the side and four to the top to get four squared. This can also be useful for finding the square of a big number quickly. For instance, the square of 52 = 50² + 50 + 51 + 51 + 52 = 2500 + 204 = 2704.

Read more at Wikipedia.org