Perfect square

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 n 2 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. a 2 ± 2 ab + b 2 = ( 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:

10 imes 10 = 10^2 - 0^2 = 100 - 0 = 100

9 imes 11 = 10^2 - 1^2 = 100 - 1 = 99

8 imes 12 = 10^2 - 2^2 = 100 - 4 = 96

7 imes 13 = 10^2 - 3^2 = 100 - 9 = 91

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

A imes B = [(A+B)/2]^2 - [(A-B)/2]^2

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

A imes B = A imes (B-1) + A

Example:

27 imes 34 = [27 imes 33] + 27 = [30^2 - 3^2] + 27 = 900 - 9 + 27 = 918

==See also==

[http://snafumedia.com/square.htm List of perfect square between 1-10,000]

[http://digitalfilipino.21publish.com/simoncpu/weblogEntry/13s26v4b3orbs.htm JavaScript code for finding ten-digit numbers, consisting of distinct digits, that are perfect squares]