Addition of complex numbers in polar form

[Collin Li]

There's a technique that I proudly discovered independently during my year 12 days. I found out a way to add complex numbers that were in polar form where was the same in all the terms.

In other words: express as one complex number.











There you have it. An addition of complex numbers in polar form. The strategy is as follows: factor out a complex number with an argument that is the midpoint of the arguments of the two complex numbers to be summed, resulting in the sum of a complex number and its conjugate. This simplifies into a real number so that it is part of the magnitude and no longer effects the argument.

You can also apply this technique for subtraction, however the residue from the subtraction of conjugates will be something like (a pure imaginary number: no real parts), which means you need to make an extra step to take into the argument.

The usefulness of this? There are some cases where it is much easier to do this, than to convert into cartesian form. You can apply this technique to efficiently prove that , where .

[cara.mel]

does r have to be the same size?

Why did they lie and say you couldnt add/subtract in polar form. My book distinctly said that.
I hate you, VCE.

[dcc]

It seems like a bit of trouble for the book to go over the specific case where lol

[midas_touch]

Demand that this gets introduced into the spesh syllabus, where it will be known as 'Coblins Law'.

[Ninox]

woo! coblin's law FTW!
coblin, you should tightly guard its secrets to third parties and non VN'ers.
Advertise "Coblin's Law" as one of your special techniques with your tutoring

[Ahmad]

Good stuff coblin!

[midas_touch]

Now its ur turn to find a new rule/law Ahmad, perhaps a working title to be called 'Ahmad's law of ownage' .

[beezy4eva]

coblin ur awesome.
and i was testing it out u can do subtraction using the addition way
say z1= r cis θ1  and z2= r cis θ2 and u want 2 find z1-z2, you make it z1 + (-1 x z2). So it becomes  r cis θ1 + r cis (θ2 ± pi)

i can prove it works using vectors but i cbf

ps sorry about the forum necromancy

[Collin Li]

For subtraction:

You can also apply this technique for subtraction, however the residue from the subtraction of conjugates will be something like (a pure imaginary number: no real parts), which means you need to make an extra step to take into the argument.

[Collin Li]

I have been questioned about the "legality" of this technique for use on SACs and exams. It is 100% legal because it does not draw upon any new theory. It is a purely algebraic manipulation, using theorems that you have been taught already (such as de Moivre's). An example of an illegal use of mathematics would be the use of double derivatives, to justify the nature of a stationary point, on a Methods exam (draws on theory that is not in the Methods course - it's certainly in the Specialist course though!).

Because it may appear unfamiliar to the examiner, perhaps you should take the steps slower, and annotate the critical steps, explaining what you are doing, and your motivation for it.

[ed_saifa]

I have been questioned about the "legality" of this technique for use on SACs and exams. It is 100% legal because it does not draw upon any new theory. It is a purely algebraic manipulation, using theorems that you have been taught already (such as de Moivre's). An example of an illegal use of mathematics would be the use of double derivatives, to justify the nature of a stationary point, on a Methods exam (draws on theory that is not in the Methods course - it's certainly in the Specialist course though!).

Because it may appear unfamiliar to the examiner, perhaps you should take the steps slower, and annotate the critical steps, explaining what you are doing, and your motivation for it.

My teacher said that it is "legal" to use specialist techniques in a methods exam unless the question specifically says to use a particular technique.

[Collin Li]

I would not do it. The theory involving double derivatives is not on the Methods course. It is not a "technique" that is in the course. You can only use this theory if you lay down the proof describing why a double derivative suggests a particular nature for the stationary point (not too hard, but still a waste of time for the examiner and you).

That said, however, the MM exam will probably never ask you to justify the nature of a stationary point because (i) this technical debate here is going to cause some contention, and (ii) even if most of the people are doing the first derivative sign test, it is highly mechanical and trivial to examine.

Another example: cross products - don't use them.

[evaporade]

Operating complex numbers in polar form is the most suitable way for multiplication, division, finding power and roots. For addition and subtraction of complex numbers, it is much easier to carry out in x + yi form.

For people wishing to waste time in adding and subtracting complex numbers in polar form, try the following stupid formulae.
[IMG]http://img204.imageshack.us/img204/9914/vcenotesforum1ek3.gif[/img]

[Collin Li]

I have already mentioned reasons why this manipulation of addition of two complex numbers of the same magnitude (in polar form) is useful.

Show that , where using a similar technique.

Or find the modulus and argument of .

The transformation (for the general case) you've given above is not so useful, however. Having cartesian form to add and subtract is nice, but sometimes it is not possible to do it nicely from polar form, or it is just slower. For this reason, exploiting the midpoint is a powerful technique in some cases.