Friday, September 14, 2007
Imaginary Numbers Really Exist (I'm Really, Really Serious!)
Warning: Geek stuff ahead.
As children we learn to do some basic counting and learn to recognize basic colors.
We consider this counting to be as "real" and "true" as we do the colors. (Remember the big, fat Crayolas in the box of 8 we all had in first grade?)
Some of us eventually learn the difficult task of using fractions while some never get the hang of it but learn to make do; my daughters still have trouble choosing the right measuring cup,
We have even learned to accept that negative numbers are real.
We have a number line, with 0 in the middle, positive numbers heading off into infinity on the right, while negative numbers do the same on the left of the zero. Now them negative numbers are a bit disconcerting to some folks, but after years of being charged huge fees by their banks because their checking accounts dipped into negative territory, even these disconcerted folks understand that a negative number can exist and also have an impact on their lives to boot.
If I'm Silas Marner, sitting alone in my humble abode counting my coins, I might be able to make do with just the number line approach to numbers and might consider fractions to be useless and even maybe that they don't exist at all.
But time and the advancing world have shown that, yes, even fractional numbers exist and in some cases are quite useful.
If Silas Marner became the person that had to compare the sizes of the piles of coins in each home in his community, fractions would be very useful. Bob's pile of coins might only be 1/2 the size of Mr. Marner's. You get the picture.
Heck, even the Greeks (the ancient ones) had number issues. For a long time they believed that all numbers were "rational," or that all numbers could be expressed as a ratio of two integers. But they finally bowed to proof that some numbers when expressed as a ratio of two integers and the division is carried out, result in an answer that goes on forever. These numbers were grudgingly accepted and called "irrational" numbers.
Pi is an irrational number; it goes on forever even if we mostly abbreviate it to be simply 3.14.
One problem is that, ALL numbers are abstracts. At the very least we all learned to count and process numbers in very basic ways. But I can show you a rock, say five rocks, but I can't show you 5. 5 is an abstraction. It's something we mentally assign to some things.
Throughout history we have slowly gained acceptance of the usefulness of numbers and the mathematics to handle those numbers, but they are still abstract concepts that we've agreed on.
In the 16th century a man named Rafael Bombelli wrote about algebra and laid the foundations for what would unfortunately become known as "imaginary numbers" and "complex numbers."
Here's the deal: we all learn that when multiplying a number with itself, that is referred to as "squaring" that number. 2*2=4 can be said as "two squared equals four." We also learned that the square of a negative number results in a positive number. -2*-2=4 is true as well.
Mr. Bombelli's work in algebra showed that it would be useful to have a number that represents the square root of -1. In other words, he proved mathematically that there are times that the square root of -1 is useful and that he believed this number to exist.
He was swimming against the tide of accepted math. EVERYBODY knew that squaring a number, whether positive or negative, resulted in a positive number, always. Period.
Years later, none other than Rene Descartes referred to these numbers as "imaginary numbers" and that name stuck. Mr. Descartes was using this in the frame of mind that we use with a child's "imagnary" friend. That friend doesn't really exist. Mr. Descartes was using the word imaginary as a derogatory term in his desire for this to just go away.
Sadly the word "imaginary" was forever attached to these numbers, though their existence has long since been proven to be every bit as "real" as irrational numbers and fractions.
Despite their still being called imaginary numbers, imaginary numbers exist and are as useful in some circles as are fractions to a bookie, or as useful as Pi is when computing the area of a circle.
Right there along with imaginary numbers are what are referred to as "complex numbers." Complex numbers are the same old numbers we've used all of our lives, that live and breathe along the number line, along with an imaginary component. It is usually written in one of two ways: 4 + 7i, or 4 + 7j, where the 7i or 7j part is the imaginary part of the complex number.
The number 4 can also be written as 4 + 0i, or 4 + 0j. Zero times any number is zero, and 4 + 0 = 4.
Still with me?
All numbers along the aforementioned number line we learn about in grade school can be expressed as complex numbers. And if you look at my spiffy hand drawn picture at the top of the post, you'll see an old-school number line with a vertical one crossing it at the number 0. The old number line part still represents the real numbers that we've all used all of our lives, but the added vertical number line represents the imaginary part of a complex number. The whole thing is called the Complex Plane.
By convention, most people dealing with complex or imaginary numbers use the lower case letter "i" to represent the square root of -1. But, in electical engineering, we use that lower case "i" to represent current in an electrical circuit.
So, while your experience with complex or imaginary numbers would normally use "i"; we use "j" in electrical engineering to avoid confusion with current.
Still with me?
In electrical engineering, when dealing with Alternating Current circuits like your favorite table lamp uses, the trigonometry functions sine and cosine come into play. The mathematical representation for the value of, say, power, at any given time is what is referred to as a "phasor." Power company engineers live and breathe this "complex power" and it accurately represents their power distribution systems.
Still with me?
I'll back off now. After this things get really hard to explain, but just understand that power utilities and the whole science and engineering involved in Alternating Current electrical systems that are used all over the world, DEPEND on the mathematics of complex and imaginary numbers to be able to deliver power to your home and to businesses.
I was dealing with it a bit at work yesterday and thought I would see if anyone out there might be interested to know that imaginary numbers truly exist.
Besides, my calculator will even compute with complex numbers, so it must be true, right?
Imaginary numbers just have an unfortunate name; kinda like being a movie or rock star's child, ya know? (I personally would rather be named Imaginary Number than to be named Dweezil or Moon Unit.)
You probably don't care a lick, but I took the time to type all of this out, and I'm posting it.