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/