Hey guys, I'm still an undergrad, but I've got a Professor that's been raving on about "imaginary" numbers and how he thinks they should be called "electrical" instead of "imaginary." I'm not even into Calculus yet, so if any of you Electrical Engineers haven't used these on the job in awhile, it may take some refreshment...

The "imaginary" numbers I'm referring to are the square roots of negative numbers, or to be more precise, the square root of negative one (r[-1]), represented by an "i" for "imaginary." ("i" squared is equal to "-1.") I can see how these numbers could be used plotting points or graphing images on an oscilloscope or similar measurement instrument, but are there other uses for them also?

And does anybody know of a relationship between two consecutive (i.e. 12 & 13, 15 & 16, 100 & 101, etc.) squares or square roots? Somebody has had to have noticed this before, but I have come up with a forumla to help find the next chronological (or previous chronological) square when one is known; it's just a quick way to find a rational number's square from a chronological sequence without using a calculator. Here's the formula and it's explanation:

(This computer doensn't have any keys that will resemble a "raised 2," meaning "squared," so I'll just use the variable times itself to represent that.)

(N x N) + N + X = (X x X) (lowercase "x's" are the symbol for multiply)
Where N = a rational number to be squared that is known (can be ANY whole number from 0 to + infinity)
And X = the next chronological number that is not known or needs to be calculated, and is equal to (N + 1).

The formula also works in reverse, using:
(N x N) - N - Y = (Y x Y) (the only difference in this formula is the variable "Y," which is one number less than "N," or equal to (N - 1), and subtraction instead of addition)

Examples:
N = 0
X = 1
(0 x 0) + 0 + 1 = (1 x 1)
0 + 1 = 1

N = 15
X = 16
(15 x 15) + 15 + 16 = (16 x 16)
(225) + 31 = (16 x 16)
256 = (16 x 16)

"Backwards" examples:
N = 1
Y = 0
(1 x 1) - 1 - 0 = (0 x 0)
1 - 1 = 0

N = 100
Y = 99
(100 x 100) - 100 - 99 = (99 x 99)
(10000) - 199 = (99 x 99)
9801 = (99 x 99)

I've done plenty of other examples too-and the formula works for all rational numbers that I've tried, but I haven't come across anything in any of my math books similar to this yet. Since squares have been around for centuries, somebody else has had to figure this out before, right? Of course it's not absolutely practical to work this formula out when you have a calculator right in front of you, but I noticed the pattern when I was trying to figure out a way to remember squares higher than most multiplication tables show.

If anybody has read or remembered anything similar to this from school, or even teaches something like this, please let me know. I'd also like to know more about the practical application of "i," or the square root of (-1). Thanks in advance for your time, help, and cooperation...

Sincerely,
-Adam Collins