Question

Complex Equation
|x + iy| = y - ix

Answer 1

ok so i get that it will split up into:
y = |x|
and
-x = |y|
right? so \[y \ge 0\]

Answer 2

but then our book says that x=0?

Answer 3

Your book is quite right :)

Answer 4

...listen up... this part on the left... (which I thoughtfully coloured in blue) is going to be a plain-and-simple real number, right? (And a positive one, at that)

Because it's an ABSOLUTE VALUE (or modulus, si vous preferez)
\[\Large \color{blue}{|x+iy|}=y - ix\]

...right?
^_^

Answer 5

yah

Answer 6

:)

Answer 7

Well then, that forces the RIGHT side of the equation (which I thoughtfully coloured in red) to also be a plain-and-simple real number:
\[\Large |x+iy|= \color{red}{y-ix}\]

Answer 8

And if the right-side is a pure real number, then x MUST be zero, otherwise, it would have an imaginary component, which basically spells the end of the world, all right? :3

Answer 9

hrm

Answer 10

why does it have to be 0 though?

Answer 11

if it isn't, then the right-side gains an imaginary part...

Answer 12

Which is impossible, because, it is equal to the absolute value of something, which may NEVER have an imaginary part.

Answer 13

Get it?
It's just as absurd as having an absolute value equal to some negative number, say
|z| = -5

It simply isn't possible ^_^

Answer 14

@ksmith197
Do you understand now? ^_^

Answer 15

hey sorry off doing other stuff :o
um yah well logically i understand what you are saying. you can't have an absolute value equal to an imaginary number

Answer 16

so the only value it can equal is 0

Answer 17

but how do you write that as a solution?

Answer 18

More or less...

more like you can't have an absolute value with an imaginary part.

Answer 19

couldn't you do a transformation of it though? z= x + iy

Answer 20

ooo i think that may be it. we have other problems that are similar, where z= x + iy = sqrt( x^2 + y^2)

Answer 21

eliminating the i

Answer 22

something like that, still, it all boils down to the fact that the left side has NO imaginary part, and therefore, so must the right-side.

Makes sense, yeah?

Answer 23

yah that makes sense. thank you very much :) :)