Suppose z1 = a + bi, z2 = c + di are complex numbers.(adsbygoogle = window.adsbygoogle || []).push({});

When does |z1 + z2| = |z1| - |z2|? (with || is modulus)

It seems obvious that this is the case when z2 = 0, but are there other solutions? According to the book, no. But after 2 days, I still cannot solve it!

Here is what I did:

[tex]\sqrt{(a+c)^2 + (b+d)^2} = \sqrt{a^2 + b^2} - \sqrt{c^2 + d^2}[/tex]

Then I squared both sides, remembering that [tex]\sqrt{a^2 + b^2} - \sqrt{c^2 + d^2} > 0[/tex]

If I work this out I get 3 conditions that need to be satisfied:

[tex]1) \sqrt{a^2 + b^2} - \sqrt{c^2 + d^2} > 0[/tex]

[tex]2) ac + bd \geq 0[/tex]

[tex]3) ad - bc = 0[/tex]

I do not see how this can be equivalent with z2 = 0, so I tried another way. Instead of squaring both sides immediately, I made both sides positive:

[tex]\sqrt{(a+c)^2 + (b+d)^2} + \sqrt{c^2 + d^2} = \sqrt{a^2 + b^2}[/tex]

When I work this out, I get 2 conditions:

[tex]1) ac + bd + c^2 + d^2 \leq 0[/tex]

[tex]2) ad - bc = 0[/tex]

I still don't see how this means that z2 = 0, furthermore I suspect that both sets of conditions need to be equivalent but I cannot prove this.

So I actually have 2 requests:

1. Can someone tell me how to solve this?

2. Can someone show me how both sets of conditions are equivalent to each other, if they are? And if they are not, did I make a mistake?

Thanks.

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# Complex numbers and modulus problem

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