Questions about applicable math in Electrical Engineering...

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

Adam,
The imaginary number i is used in Electrical Engineering just as a tool when dealing with signal phases. i is very useful specially when you want to calculate the power in a system where the current I and the voltage V are out of phase. Remember, i doesn't have a real meaning it's just a mathematical tool.
Now, the formula that you provided it is just a simple manipulation of numbers. If you substract N from both sides and simplify you would end up with X*N=X*N. It is just a simple algebra manipulation but it is good you caught it.

Thanks for the info on the "imaginary" numbers, I really appreciate it.

We may have misinterpreted our view on the formula I had posted, but I'm sure that was my fault due to disorganized and underinforming you guys of the patterns I had noticed when dealing with consecutive/chronological/incremental (by one) of the values of both "N" and "X" (or "Y"). Let me start by introducting the patterns I had discovered:

"N1" = 0, (0 x 0) or (0 squared) = 0
"N2" = 1, (1 x 1) or (1 squared) = 1
The above two numbers, after being squared, have a difference of "1," indicating that the SUM of both "N1" and "N2" before being squared are equal to "1", the figure that is equal to the sum (or difference in this example) of both of the numbers after being squared. (This is the only time this pattern occurs, the exact SUM and DIFFERENCE of the original numbers before squaring being equal to the SUM and DIFFERENCE of the numbers after squaring.)

"N1" = 1, (1 x 1) or (1 squared) = 1
"N2" = 2, (2 x 2) or (2 squared) = 4
The above two numbers, after being squared, have a difference of "3," (4 - 1 = 3, or "N2" - "N1" = 3) indicating that the SUM of both "N1" and "N2" BEFORE being squared are equal to "3," the figure that is equal to the DIFFERENCE of "N2" and "N1" ("N2" - "N1" = 3) after both numbers have been squared. Notice a pattern yet? Here it is: THE DIFFERENCE OF THE TWO CONSECUTIVE NUMBERS SQUARED IS ONLY TWO NUMBERS DIFFERENT (+ 2) COMPARED TO THE PREVIOUS EXAMPLE (which occured in a chronological order between the first and second examples, i.e., example one had "N1" = 0 and "N2" = 1 before (and after) being squared-the only two numbers that will give the exact original number both before and after squaring, THEN example two-before being squared-used "N2" from example one as "N1" in this new example, with "N1" now being equal to 1 and a new constant for "N2," being ("N1" + 1) the number 2. Thier DIFFERENCE after both numbers were squared? 3. This had shown me that the DIFFERENCE between the two consecutive squares was equal to their original sums before squaring. A few more to help you guys understand the underlying pattern here...

"N1" = 2, (2 x 2) or (2 squared) = 4
"N2" = 3, (3 x 3) or (3 squared) = 9
Once again, the two numbers' DIFFERENCE ("N2" - "N1" = 5) after squaring is equal to the SUM of the original numbers before being squared. Can you spot the UNDERLYING pattern yet? Here's a clue: Example 1 = (0 + 1 = 1), Example 2 = (1 + 2 = 3), and Example 3 = (2 + 3 = 5). Here's ONE more example before I reveal the pattern I had mentioned...

"N1" = 3, (3 x 3) or (3 squared) = 9
"N2" = 4, (4 x 4) or (4 squared) = 16
Yet again, the DIFFERENCE of the two numbers AFTER being squared ("N2" - "N1" = 7) is equal to the SUM of the two original numbers BEFORE being squarred (["N1," or 3] + ["N2," or 4] = 7). If you haven't already noticed it, each example continues to prove that the SUM of the original numbers-BEFORE being squared is equal to the DIFFERENCE of the two numbers after being squared. Each and EVERY TIME this occurs with consecutive or chronological rational (whole) numbers, the difference between examples is equal to two (2).

I cannot claim credit for this until I can confirm that it has NOT been mentioned or discovered before; that's why I'm asking you guys-probably the most highly educated group of people on any forum I've found (don't blush-you guys deserve it for all your hard work). That's when I came up with the formula-after discovering this pattern.

0 x 0 = 0 (Reference point)
1 x 1 = 1 (0 + 1 = 1) (1 - 0 = 1)
2 x 2 = 4 (1 + 2 = 3) (4 - 1 = 3)
3 x 3 = 9 (2 + 3 = 5) (9 - 4 = 5)
4 x 4 = 16 (3 + 4 = 7) (16 - 9 = 7)
5 x 5 = 25 (4 + 5 = 9) (25 - 16 = 9)
6 x 6 = 36 (5 + 6 = 11) (36 - 25 = 11)
7 x 7 = 49 (6 + 7 = 13) (49 - 36 = 13)
8 x 8 = 64 (7 + 8 = 15) (64 - 49 = 15)
9 x 9 = 81 (8 + 9 = 17) (81 - 64 = 17)
10 x 10 = 100 (9 + 10 = 19) (100 - 81 = 19)
11 x 11 = 121 (10 + 11 = 21) (121 - 100 = 21)
12 x 12 = 144 (11 + 12 = 23) (144 - 121 = 23)

Look at the numbers on the far right hand side of each parenthetical equation beside the squared numbers above-going down the list, each of the numbers immediately beside the ")" on both equations representing consecutive or chronological numbers continue to increase by two (2) as you read down. Reading up, the numbers in the same location continue to decrease by two (2). This realization told me that a simple equation could be used with the number preceeding or proceeding ANY selected number to determine it's square, based on that information alone...

I'd be thrilled if nobody else had figured this out and had come up with an equation similar to mine or one that provided and prove the same relationship, but I'm almost sure it's been done before-even though I'd never heard of it until I had attempted to learn a method to help me remember larger squares and identify the square root of larger numbers without the use of a calculator...

Does this get the basic point across, RSOCOR01? I'm sure I (or anybody else) could revise and further simplify the formula/equation, but I had just posted what I had (believed to have) discovered at the time. I've got a formula to help found squares for ANY number, no matter how far away, based upon only the knowledge of a single rational number and it's square, and of course the number you need to know or find the square of. I'll try to revise it and simplify it as much as possible before posting it here, but it may be a few days away. I'll get behind in classes if I don't bust my ass within the next 30 hours to finish a pile of ill-defined problems that I have to have completed and turned in Tuesday morning...

BTW-Thanks for everybody's time, help, and cooperation/consideration on this matter as I attempt to research it further in the small window of free time I do have at the present moment...I greatly appreciate your help...

Sincerely,

Adam,
It looks like you really enjoy Number Theory. What you are talking about is well documented and people have been working with squares since the beginning of civilization (remember the Pithagorean theorem). Get a book about the history of Math or Number theory, you will enjoy it.
The pattern that you are talking about is called "The Difference of Squares". If you have two whole numbers X and Y then
(X^2) - (Y^2) = (X-Y)(X+Y)
If you have two consecutive numbers then (X-Y)=1, so the difference would be the sum of the numbers. Hope that this helps.

Robert

Thanks for the info-I knew that somebody HAD to have noticed that before, but I had figured the equation out before being shown. There are plenty of ways to revise that formula to find squares many numbers away, but it's really more efficient to just grab a calculator (or even to just write out the number to be squared)...

I'll have to search for the "Number Theory" when I get another chunk of homework knocked out. I really appreciate your feedback, rsocor01. It's the many educated and skilled members of this forum like yourself that make it such a great place. Sorry if it had seemed like I was irritated or a bit harsh in that last post (I had just read it again since I had posted it and I didn't particularly like the way I had it written), I was just up late and aggravated with my homework; I didn't mean to come off like an ass, so please don't take it that way.

I see where you had gotten "X*N=X*N" now after working a few examples with your last equation, (X^2) - (Y^2) = (X-Y)(X+Y). I know how to use "F.O.I.L." and factor, but the only "difference of squares" I remember learning was noticing the two identical variables with opposite signs and knowing that the outer and inner multiplication would cancel each other out. I wish my teachers would have gone into more depth on that...

Thanks for your time and patience with me...