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### Why Multiply Two Complex Numbers?

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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?

Thanks.
```

```
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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Imaginary/Complex Numbers

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