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Why Multiply Two Complex Numbers?

Date: 02/20/99 at 17:14:08
From: Angkit Panda
Subject: Multiplication of Complex Numbers in Form of Vectors?

Why do we multiply two complex numbers?  What does that give us, and 
how do you graph it?  

	(8 cis 70) * (2 cis 34) = (16 cis 104)

	How do you get that?  What is it used for?


Date: 02/22/99 at 09:00:41
From: Doctor Peterson
Subject: Re: Multiplication of Complex Numbers in Form of Vectors?

The basic idea behind multipliying complex numbers is a simple 
application of the distributive property:

    (a + bi)*(c + di) = ac + adi + bci + bdi^2
                      = (ac - bd) + (ad + bc)i

Someone discovered that this fits very neatly with the polar ("cis") 
representation of the numbers:

 (a cis b)*(c cis d) = (a cos(b) + a i sin(b))*(c cos(d) + c i sin(d))
                     = [ac cos(b) cos(d) - ac sin(b) sin(d)] +
                       [ac sin(b) cos(d) + ac cos(b) sin(d)]i
                     = ac cos(b+d) + ac sin(b+d)
                     = ac cis (b+d)

So this method of multiplication of complex numbers comes directly from 
the angle-sum identities in trigonometry. This gives us a simple way to 
see complex multiplication in terms of vectors: multiply the lengths 
and add the angles. But that does not originate in any vector concept; 
it is simply the natural result of defining multiplication of complex 
numbers to follow the same rules as real numbers. Addition of vectors 
is a natural concept; multiplication applies only to complex numbers.

Write back if there is something else you need to understand.

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Imaginary/Complex Numbers

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