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? 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/
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