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If z1 and z2 are two non-zero complex numbers such that mod(z1+z2)=mod(z1)+mod(z2) then arg(z1)-arg(z2)=?
Now when I see this question, I think it must be -pi because I know only that much. I can't actually visualize the question,
so please help me on understanding the concept of argument along with some special tricks and techniques for finding the
arguments,(it may be some figures,graphs etc.) Thanks in advance!!!
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On complex numbers, there are introductory lessons in this site
- Complex numbers and arithmetical operations on them
- Complex plane
- Addition and subtraction of complex numbers in complex plane
- Multiplication and division of complex numbers in complex plane
Read them, and you will know EVERYTHING about complex numbers what is required to understand your problem and solve it.
You will learn that
"If z1 and z2 are two non-zero complex numbers such that mod(z1+z2) = mod(z1) + mod(z2)"
then it may happen it and only if the numbers z1 and z2 lie in one ray released from the origin of the coordinate system (of the complex plane).
Why ? - Because adding of complex numbers is the same as adding vectors representing these complex numbers, and, as such,
adding the complex numbers obeys "the parallelogram rule".
And then you will get the understanding that "mod(z1+z2) = mod(z1) + mod(z2)" for complex numbers implies that
arg(z1) - arg(z2) = 0,
or, in the most general case,
arg(z1) - arg(z2) = ,
where k is any integer k = 0, +/-1, +/-2, . . .
Again: read the referred lessons.
They are written by me specially for you.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic "Complex numbers".