Definition

# 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
Posted by: Margaret Rouse

#### Email Alerts

Register now to receive SearchCIO-MidMarket.com-related news, tips and more, delivered to your inbox.
By submitting you agree to receive email from TechTarget and its partners. If you reside outside of the United States, you consent to having your personal data transferred to and processed in the United States. Privacy

#### More News and Tutorials

• ##### Remote backup can ease network disaster recovery

Backing up to local disk or tape can protect against the accidental destruction of a file but will do nothing to protect you when your facility suffers fire or flooding. Remote backup provides a better solution. This tip looks at two possibilities for remote backup: Choosing an outsourced remote backup service, or using software to do your own backups to an offsite facility.

• ##### VoIP now part of phishing attacks

Learn how attackers are using the widespread deployment of low-cost VoIP to leverage phishing attacks.

• ##### VoIP privacy on the WAN

Is the government listening to your VoIP traffic? Can other companies spy on your WAN? It may sound like a paranoid conspiracy theory, but it's not that far out, technologically speaking. In this tip, Tom Lancaster looks at just how exposed your WAN traffic may be.

#### Do you have something to add to this definition? Let us know.

Send your comments to techterms@whatis.com

#### Contribute to the conversation

All fields are required. Comments will appear at the bottom of the article.