Operations and Complex Numbers
Date: 12/04/2001 at 18:24:17 From: Matt Subject: Complex and imaginary numbers Hi, I am doing a report on complex and imaginary numbers for my 8th grade course. I understand the concept of i but I have three questions: 1) How does one do the standard operations such as addition and multiplication? 2) Why was "i" invented and what are its real life uses? 3) What exactly is a complex number?
Date: 12/05/2001 at 12:20:36 From: Doctor Mitteldorf Subject: Re: Complex and imaginary numbers Dear Matt, Books have been written on these questions. I'll give you some short answers. 1) Adding: just keep track of the real part and the imaginary part, and add them separately. For example 1+2i can be added to 3+4i and the answer is 4+6i. Multiplying: Use the distributive rule and the rules for multiplication that you know. To multiply 1+2i times 3+4i you'd get 4 terms: 1*3 + 1*4i + 2i*3 * 2i*4i. You already know how to do each of these terms separately: 1*3 = 3; 1*4i = 4i; 2i*3 = 6i; 2i*4i = 8*i*i = -8. The only tricky one was the last one, where you have to remember that i*i = -1. Now add up the four terms. Two of them are real, and two have i's in them. The result is -5+10i. 2) i was invented when someone said "We can't solve equations like x^2+1=0. But what if we could? We don't know anything about the answer, but we'll just call the it "i" and see where it takes us. The interesting thing he found was that he could then solve ANY equation, not just square roots of negative numbers. For example, x^4 -9x^2 = 7. Any "polynomial" equation that takes the form of a sum of terms with x, x^2, x^3, etc. can be solved with complex numbers (real and imaginary); but it often can't be solved if only real numbers are used. Here's a good exercise for you, if you know a little algebra. What is the square root of i? From what I just said, it must be a complex number, because it is a solution to the polynomial equation x^2 = i. So let's assume there's an answer x, and that it is a complex number a+bi. See if you can find a and b. Here's how: multiply a+bi by itself. There will be a real part and an imaginary part. Write an equation that says the imaginary part is just plain i, and another equation that says the real part is zero. Solve these two equations and you'll know a and b. Then check to see if this crazy scheme really worked. Take your number a+bi, multiply it by itself, and see if you really do get i. 2a) Real life uses: It turns out that complex numbers are very useful in physics, especially in calculations about waves. The reason for that is too hard to explain for now. But one thing I'll say is that when you're all done with your physics calculation and you get an answer like "this is how loud the sound is" or "this is how fast the wave travels," those answers must always be real numbers. It's for the calculations along the way that things can be easier if you use complex numbers, even though all the i's have to cancel out in the end. 3) "What exactly is a complex number?" That depends on what you mean by the word "is." You know how to write a complex number as a real part and an imaginary part added together. You know how to add and subtract and multiply complex numbers. (Perhaps you can surmise how to divide them.) Someday you'll learn how to solve real problems in the real world using complex numbers. Maybe that's all there is to complex numbers, and there isn't any more "is" than that. I hope you'll read more about complex numbers right here at the Math Forum. You can start with this page from the Dr. Math FAQ: Imaginary Numbers http://www.mathforum.org/dr.math/faq/faq.imag.num.html - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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