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 = i
2i2 = (-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 July 2000
Dig Deeper
-
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.
-
Learn how attackers are using the widespread deployment of low-cost VoIP to leverage phishing attacks.
-
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.
-
People who read this also read...