# What's the formula for adding two complex numbers in polar form?

## Main Question or Discussion Point

I really, really need to know the formula that adds (or subtracts) two complex numbers in polar form, and NOT in rectangular form. I know there is such formula (I saw it in some book), and it's composed of cosines and sines.

Please, please don't tell me to convert back to rectangular form, add them, then covert them back to polar form--that's not what I'm looking for. I'm looking for a formula that accepts two complex numbers in polar form, adds them, and gives the answer in polar form.

Thank you.

berkeman
Mentor
I really, really need to know the formula that adds (or subtracts) two complex numbers in polar form, and NOT in rectangular form. I know there is such formula (I saw it in some book), and it's composed of cosines and sines.

Please, please don't tell me to convert back to rectangular form, add them, then covert them back to polar form--that's not what I'm looking for. I'm looking for a formula that accepts two complex numbers in polar form, adds them, and gives the answer in polar form.

Thank you.
Well, what do you think it would look like? It would have to use something like the law of sines or cosines, wouldn't it?

It would have to use something like the law of sines or cosines, wouldn't it?

berkeman
Mentor
Then why don't you just figure it out?

Then why don't you just figure it out?
I CAN'T. I've tried, but I can't really. Also, I have to use it in an exam tomorrow, and I've got other stuff to study, and I don't have time to figure it out. Please.

berkeman
Mentor
I CAN'T. I've tried, but I can't really. Also, I have to use it in an exam tomorrow, and I've got other stuff to study, and I don't have time to figure it out. Please.
But you can use a search engine, right? It took me about 30 seconds to find it with a google search. I googled vector addition polar coordinates, and got lots of hits. One of the places it suggested was Hyperphyics:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Go there, and search on vector addition. That will get you to the formula.

I'm not so much trying to be a jerk here -- you need to learn how to learn, in addition to learning the material. Hope the test goes well.

Mentallic
Homework Helper
But if you convert to rectangular form, add, then convert back to polar form, won't that give you the formula you're asking for?

But you can use a search engine, right? It took me about 30 seconds to find it with a google search. I googled vector addition polar coordinates, and got lots of hits. One of the places it suggested was Hyperphyics:

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Go there, and search on vector addition. That will get you to the formula.

I'm not so much trying to be a jerk here -- you need to learn how to learn, in addition to learning the material. Hope the test goes well.
I've used a search engine many times before I posted this question here. I couldn't find anything. I've a faint idea that the formula has to do with parallelogram law. However, I need a formula for adding two complex numbers in polar form, so the vectors have to be in polar form as well.

Mentallic -- I've tried your idea, but there are two parts of the complex number to consider--the real and the imaginary part. I just can't figure how to get them. I'll try some more. If I get the formula I'll post it here.

berkeman
Mentor
I've used a search engine many times before I posted this question here. I couldn't find anything. I've a faint idea that the formula has to do with parallelogram law. However, I need a formula for adding two complex numbers in polar form, so the vectors have to be in polar form as well.

Mentallic -- I've tried your idea, but there are two parts of the complex number to consider--the real and the imaginary part. I just can't figure how to get them. I'll try some more. If I get the formula I'll post it here.
Treat the complex numbers like vectors to do the addition (and to figure out the formula that you want). The Real part is just the part in the x-direction, and the Imaginary part is in the y-direction.

As Mentallic says, just write out the sum in components, and convert back to polar as part of the whole equation.

$$\textbf{A} = A < \theta_1$$

$$\textbf{B} = B < \theta_2$$

$$\textbf{A} + \textbf{B} = ?$$

Please, please don't tell me to convert back to rectangular form, add them, then covert them back to polar form--that's not what I'm looking for.
Convert them back to rectangular form, add them, convert them back.

Seriously. What are you getting worked up over? That's the only way to do it. Any other way is going to be equivalent to that.

To be technical, the "numbers" themselves don't have forms. "Polar form" and "rectangular form" are two different representations of those numbers. They aren't the only two representations either. You could easily covert complex numbers to matrix form, then just use matrix multiplication, then convert to polar form again.

Regardless of how you try to do it, your formula will just be a disguised variation of converting the numbers to and from rectangular form.

eumyang
Homework Helper
I really, really need to know the formula that adds (or subtracts) two complex numbers in polar form, and NOT in rectangular form. I know there is such formula (I saw it in some book), and it's composed of cosines and sines.
Are you sure you saw it in some book? I may be looking at the wrong books, but I've checked a number of Pre-Calculus books and none of them had any formula for adding/subtracting complex numbers in polar form. They did have formulas for multiplying/dividing complex numbers in polar form, DeMoivre's Theorem, and roots of complex numbers. I'm not trying to be a jerk here, either, but I'm wondering if you're confusing formulas.

69

OK, for a complex number a, written as r ei(theta), what is that in rectangular?

HallsofIvy