Evaluating complex multiplication?

In summary, complex numbers can be represented as R(& theta) where R is the magnitude and & theta is the phase. The product of two complex numbers can be represented as R*P(& theta + & Theta). To convert a complex number into R(& theta) form, use the formula r=(a^2 + b^2)^(1/2) and tan θ = b/a. Make sure to account for the correct quadrant when finding θ. There is no simplifying identity for the arithmetic or geometric series involving complex numbers.
  • #1
Loren Booda
3,125
4
How might one comprehend the product of complex numbers

N
[pi](an+ibn)=C
n=1

such as by representation with vectors in the complex plane, or algebraic simplification? Specifically, I would like to know the value Re(C).
 
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  • #2
Hi

Any complex number a+ib can be represented as R(& theta). Where R is the magnitude and & theta is the phase. So convert the complex number into R(& theta) form and multiply. The R parts multiply while the angle parts add up.

e.g. R(& theta)* P(& Theta) = R*P(& theta + & Theta)

Got it?


Sridhar
 
  • #3
A bit rough, sridhar, but helpful in jogging my memory.

Can you or another be more mathematical in regard to the transform involved?

Is it tan-1(b/a)=[the] and r=(a2+b2)1/2?
 
  • #4
Right...ish

if (a + bi) = r exp(iθ), then it is true that

r = (a^2 + b^2)^(1/2)
and
tan θ = b/a

But you have to make sure that θ is in the correct quadrant. (iow you might have to add π).
 
  • #5
Hurkyl,

Is there a simplifying (exact) identity for the arithmetic series

N
[sum]tan-1(bn/an)
n=1

and for the geometric series

N
[pi](an2+bn2)1/2
n=1

or, more importantly, for my original statement concerning Re(C)?
 

1. How do you evaluate complex multiplication?

Evaluating complex multiplication involves multiplying two complex numbers together using the rules of complex numbers. This includes distributing and combining like terms, and simplifying the resulting expression.

2. What are the rules for multiplying complex numbers?

The rules for multiplying complex numbers are as follows: (a+bi)(c+di) = ac + adi + bci + bdi^2. This can be simplified to (ac-bd) + (ad+bc)i.

3. Can you give an example of evaluating complex multiplication?

Yes, an example of evaluating complex multiplication would be (3+2i)(4-5i). Following the rules, this would simplify to (12-15) + (8-10)i, or -3-2i.

4. What is the difference between complex multiplication and regular multiplication?

Complex multiplication involves multiplying numbers that have both a real and imaginary component, while regular multiplication only involves multiplying real numbers. Complex multiplication also follows different rules and involves simplifying the resulting expression.

5. Why is it important to evaluate complex multiplication?

Evaluating complex multiplication allows us to perform mathematical operations involving complex numbers, which are often used in real-world applications such as engineering, physics, and electronics. It also helps us understand the behavior of complex numbers and their relationships with each other.

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