An imaginary number is a quantity of the form ix, where x is a real number and i is the positive square root of -1. The term "imaginary" probably originated from the fact that there is no real number z that satisfies the equation z2 = -1. But imaginary numbers are no less "real" than real numbers. The quantity i is called the unit imaginary number. In engineering, it is denoted j, and is known as the j operator.
The unit imaginary number has some intriguing properties. For example:
(-i)2 = -1
but -i is different from i
i3 = i2i = (-1)i = -i
i4 = i2i2 = (-1)(-1) = 1
i5 = i3i2 = (i3)(-1) = (-i)(-1) = i
in = i(n-4)
when n is a natural number larger than 4
As i is raised to higher natural-number powers, the resultant cycles through four values: i, -1, -i, and 1 in that order. No real number behaves like that!
The set I of imaginary numbers consists of the set of all possible products iw, where w is an element of the set R of real numbers. Therefore, the sets I and R are in one-to-one correspondence. The sum v + iw of a real number v and an imaginary number iw forms a complex number. The set C of all complex numbers corresponds one-to-one with the set R ? R of all ordered pairs of real numbers. The set C also corresponds one-to-one with the points on a geometric plane.
Imaginary and complex numbers are used in engineering, particularly in electronics. Real numbers denote electrical resistance, imaginary numbers denote reactance, and complex numbers denote impedance.