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 *z*^{2} = -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*

*i*^{3} = *i*^{2}*i* = (-1)*i* = -*i*

*i*^{4} = *i*^{2}*i*^{2} = (-1)(-1) = 1

*i*^{5} = *i*^{3}*i*^{2} = (*i*^{3})(-1) = (-*i*)(-1) = *i*

*i ^{n}* =

*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

*of real numbers. Therefore, the sets*

**R***and*

**I***are in one-to-one correspondence. The sum*

**R***v*+

*iw*of a real number

*v*and an imaginary number

*iw*forms a complex number. The set

*of all complex numbers corresponds one-to-one with the set*

**C****of all ordered pairs of real numbers. The set**

*R*?*R**also corresponds one-to-one with the points on a geometric plane.*

**C**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

*Posted by:*Margaret Rouse

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