Prove |z+w|<=|z|+|w| (1 Viewer)

[member 6003]

you have to start with this:
where z, w are any complex numbers

And prove:

My method:
let Z,W be conjugates of z,w

- expand RHS

then if the thing is true then:

square both sides:


and then I'm stuck. I saw another guy do the question on stack exchange and they basically got to the same step but then squared both sides again (letting z=a+ib, w=x+iy) and whatever, then factorise so that its like x^2>=0 therefore must be true.
But I don't understand how you can square both sides at this step since LHS can be negative, and RHS must be positive. I thought you could only square both sides of an inequality when both sides are positive.
This question is out of extension 2 Cambridge 1D question Q25 b)

 

[idkkdi]

you have to start with this:
where z, w are any complex numbers
View attachment 36399
And prove:
View attachment 36400
My method:
let Z,W be conjugates of z,w
View attachment 36399
View attachment 36402 - expand RHS
View attachment 36403
then if the thing is true then:
View attachment 36407
square both sides:
View attachment 36405
View attachment 36406
and then I'm stuck. I saw another guy do the question on stack exchange and they basically got to the same step but then squared both sides again (letting z=a+ib, w=x+iy) and whatever, then factorise so that its like x^2>=0 therefore must be true.
But I don't understand how you can square both sides at this step since LHS can be negative, and RHS must be positive. I thought you could only square both sides of an inequality when both sides are positive.
This question is out of extension 2 Cambridge 1D question Q25 b)

Square finish the proof then write it backwards and it should be legit.

 

[Trebla]

A corollary you need first is that for any complex number z (something you can easily prove):


Notice that since the terms of the sum are conjugates of each other:

 

[member 6003]

A corollary you need first is that for any complex number z (something you can easily prove):


Notice that since the terms of the sum are conjugates of each other:

thank you very sir