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# imaginary number

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.

This was last updated in September 2005

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