complex problem (1 Viewer)

[stag_j]

i have the following problem for uni and its bothering me because it seems a little too easy. is it correct to assume that since all the points lie in one quadrant and are non-zero, their sum can't possibly be 0?

suppose that compelx numbers z₁, z₂, ... , z_n lie strictly on one side of some straight line through the origin, in the complex plane.
a) show that z₁ + z₂ + ... + z_n != 0
b) Show that 1/z₁, 1/z₂, ... , 1/z_n all lie strictly on one side of a straight line through the origin.

 

[Affinity]

a.)z= x+iy

x>ky or x<ky for some k for all x+iy representing z.

so add them all up ie x_1 + x_2 + ... and y_1 + y_2 + ...
and you have sum of x > k* sum of y or the other way around, so it cannot be 0.

b.) hmm 1/z will rotate z by Pi radians about the origin.. so.. the points go to the other side of the line

 

[Archman]

Originally posted by Affinity
b.) hmm 1/z will rotate z by Pi radians about the origin.. so.. the points go to the other side of the line

err no i think 1/(rcisa) = (1/r)cis(-a)
so say the line makes an angle of b with the x-axis, so b<arg(z_k)<b+pi
so -b<arg(1/z)<-b-pi
which means all 1/z lie on one side of the line which is the reflection of the original about the x-axis.

 

[KeypadSDM]

That is so cheap.

This asessment counts, you can't just get people to tell you the answers.

More to the point, I'M ALSO DOING THIS ASSESSMENT.

 

[maniacguy]

Originally posted by KeypadSDM
That is so cheap.

This asessment counts, you can't just get people to tell you the answers.

More to the point, I'M ALSO DOING THIS ASSESSMENT.

It was also asked in a 4u half-yearly at James Ruse around 1996 (maybe a bit later than that - definitely before 2000).

 

[stag_j]

KeypadSDM,
Just for the record, I had already done the question two different ways before I posted the question. I was curious about the ways other people approach it considering i'm sure there must be many different ways of doing it.
And also, after starting this thread, I found one in the sydney uni thread discussing the question - one that even you have posted in.
If you read the junior maths handbook you'll see that they encourage students to discuss the problems together - is this any different to that?

 

[KeypadSDM]

Discussing and telling people the solutions are 2 different things.

However, the thing with maths is, once you tell someone your idea, you've effectively given them the answer.

More to the point, why didn't you just wait until the assignment was handed in to ask the question?