Imaginary Numbers.

Imaginary Numbers. Imaginary Numbers, numbers formed by multiplying a real number times i, where i is the square root of minus 1. The square root of any negative number can be expressed using i. The square root of -16, for example, is equal to i times the square root of 16 (i ): = Á  = i Certain equations, such as x2 = -1, have no real number solutions. There simply is no real number that can be substituted for x in order to fulfill this equation because the square of any real number, positive or negative, is always positive. During the 16th century mathematicians invented the concept of a whole new set of numbers, the imaginary numbers, to deal with such equations. Any sum of a real number and an imaginary number, such as 3.2 + 2i, is a complex number. More generally, complex numbers are all of the form a + bi, where a and b are real numbers but bi (the product of a real number and i) is an imaginary number. Complex numbers include all real numbers because a + bi = a when b is equal to zero. Complex numbers also include all imaginary numbers because a + bi = bi (an imaginary number) when a is equal to zero. See also Number (mathematics). Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.