I am currently doing an online self learn course and seem to have come to a bump in the exercises.
Here is the question.
Indicate if the statement is True or False. If true prove it, if false give a counter example.
arg z1z2 = arg z1 + arg z2 if z1,z2 is not equal to 0.
I have never encountered 'arg' before and have no idea what to make of it. I am currently googling away but some explanation would be helpful. No need to help me solve this, just an explanation of what 'arg z1z2' is would mean a lot!. I can't even find any reference to this in the given book or study guide.
The notation means the argument of the product of two complex numbers.
Argument is defined as the angle with the horizontal axis, and so the assertion that is saying that when you multiply two complex numbers, the result is basically a rotation consisting of the sum of the angles.
Just so I know I understand this fully. arg z1*z2 is like saying 2CIS(40) * 2CIS(22) = 1CIS(30) + 2CIS(33) . Not saying the statement is mathematically correct but that is what the question is implying?
Also how do you add in the terms like the way you guys are doing it?
I'm afraid I didn't follow. What did you mean with 2CIS(40) * 2CIS(22) = 1CIS(30) + 2CIS(33)? Let's take some specific examples: if and , we have that:
We see that or 120 degrees. Notice that and , and so .
The underlying thought is this: a complex number can be represented by a vector starting at the origin, and thus he has an angle within . When you multiply two complex numbers, the result will always sum the angles and their magnitudes will be multiplied.
In our case, if instead of and we had and , the result would be , showing that magnitudes multiplied, however the sum is unchanged.
I hope this clarified it further.