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$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$.

I approached this by letting $z=a+ib$, and $w=c+id$, and kinda play around with it. I also tried to square both sides when proving forward direction, but I could not get it to work. Can anyone get me some ideas, maybe?

Thank you!

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    $\begingroup$ $z=sw$? what's $s$? $\endgroup$ – Alex Jan 26 '14 at 19:13
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    $\begingroup$ Squaring is not a bad idea. How did you prove the triangle inequality? Looking at that proof, a condition for equality should offer itself. $\endgroup$ – Daniel Fischer Jan 26 '14 at 19:14
  • $\begingroup$ @Alex: oops, I forgot, real positive number :) $\endgroup$ – Akaichan Jan 26 '14 at 19:17
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First try to prove it in one direction.

Let $z = a + bi$ and $w = sa + sbi$. Then we have:

$$\mid z + w \mid = \mid a + bi + sa + sbi \mid = \mid |a(1+s) + (1+s)bi\mid = \sqrt{(1+s)^2a^2 + (1+s)^2b^2} = (1+s)\sqrt{a^2 + b^2} = \sqrt{a^2 + b^2} + \sqrt{s^2a^2 + s^2b^2} = \mid z \mid + \mid w \mid$$

Now for the other direction. Let $z=a + bi$ and $w = c + di$. Then we have:

$$\mid z + w \mid = \mid a + bi + c + di \mid = \mid (a+c) + (b+d)i \mid = \sqrt{(a+c)^2 + (b+d)^2}$$

$$\mid z \mid + \mid w \mid = \mid a + bi \mid + \mid c + di \mid = \sqrt{a^2 + b^2} + \sqrt{c^2 + d^2}$$

Now since $\mid z + w \mid = \mid z \mid + \mid w \mid$ we have:

$$\sqrt{a^2 + c^2 + 2ac + b^2 + d^2 + 2bd} = \sqrt{a^2 + b^2} + \sqrt{c^2 + d^2}$$

Square both sides and we have:

$$a^2 + c^2 + 2ac + b^2 + d^2 + 2bd = a^2 + b^2 + c^2 + d^2 + 2\sqrt{(a^2 + b^2)(c^2 + d^2)}$$

$$ac + bd = \sqrt{(a^2 + b^2)(c^2 + d^2)}$$

Now square again and multiply out:

$$a^2c^2 + b^2d^2 + 2abcd = a^2c^2 + b^2c^2 + a^2d^2 + b^2d^2$$ $$2abcd = b^2c^2 + a^2d^2$$ $$b^2c^2 + a^2d^2 - 2abcd = 0$$ $$(cb - ad)^2 = 0 \implies cb = ad$$

So let $c = sa$, then we have:

$$scb = sad \implies sb = d$$

Hence $z = a + bi$ and $w = c + di = as + sbi$. Q.E.D.

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  • $\begingroup$ I'm still working on it, thank you though! $\endgroup$ – Akaichan Jan 26 '14 at 19:27
  • $\begingroup$ I think I got it! $\endgroup$ – Akaichan Jan 26 '14 at 19:30
  • $\begingroup$ @Akaichan I've posted a solution for the other direction too. Hope it helps. $\endgroup$ – Stefan4024 Jan 26 '14 at 19:32
  • $\begingroup$ Thanks! I got to the $cb=ad$ part, and was stuck for a bit. $\endgroup$ – Akaichan Jan 26 '14 at 19:55

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