The term perfect square is used in mathematics in two meanings:

• a positive integer which is the square of some other integer, i.e. can be written in the form n² for some integer n.
  • Examples: 1, 4, 9, 16, 25, 36, 49, ... See square number.
• an algebraic expression that can be factored as the square of some other expression, e.g. a² ± 2ab + b² = (a ± b)². (see Square (algebra)).

This is not the same as a magic square.

Using differences of squares as multiplication

Integer multiplication can be done entirely by a difference of two squares.

Examples:

In general, the product of two numbers is equal to the square of their average minus their difference from the average squared.

A geometric constructive "proof" of this relation is shown the following animation:

The starting rectangle is A by B. The resulting large square is length (A+B)/2, and the smaller gray square (remainder being subtracted) is length |A-B|/2.

Using this relation, you can multiply relatively large nearly equal numbers more quickly if you memorize a relatively small list of squares.

If you're multiplying an even by an odd, you can avoid "halves" by adjust one number, by requiring one more addition at the end

Example:

See also

List of perfect square between 1-10,000