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# Prove that |z+w| is less than or equal to (|z| + |w|) ? Watch

1. I'm trying to understand how to prove this triangle inequality (khan academy has a video on Cauchy Schwarz Theorem...that seems more complex than this).

How do I prove it geometrically?

The book I'm using has an algebraic version, where they square both sides...then try and prove (ac + bd) is less than or equal to |z| |w|

(btw z & w are complex numbers, z=a+bi & w=c+di)
I just don't understand what the book is doing.

It has a couple lines like this:
|z||w| = sqrt\[(ac + bd)^2 + (ad-bc)^2]
|z||w| is GREATER THAN OR EQUAL TO (how the..!) sqrt\[(ac+bd)^2]
|z||w| = |ac + bd|

I don't know where the (ad-bc)^2 went or why the signs changed and changed back again...or why they're even doing it the way they are. Other explanations seem more longwinded but don't skip steps out. How would I 'prove' the triangle inequality in an exam btw?
2. Another qu is to prove that:
|z-w| is greater than or equal to (|z| - |w|)
Although I think I can figure that out if I understand how to figure out the inequality shown above..

Edit: Here is yet another different version to prove the above...I don't get this one either :/
http://www.math.ucdavis.edu/~forehan...inequality.pdf
3. (Original post by PhysicsGal)
I'm trying to understand how to prove this triangle inequality (khan academy has a video on Cauchy Schwarz Theorem...that seems more complex than this).

How do I prove it geometrically?

The book I'm using has an algebraic version, where they square both sides...then try and prove (ac + bd) is less than or equal to |z| |w|

(btw z & w are complex numbers, z=a+bi & w=c+di)
I just don't understand what the book is doing.

It has a couple lines like this:
|z||w| = sqrt\[(ac + bd)^2 + (ad-bc)^2]
|z||w| is GREATER THAN OR EQUAL TO (how the..!) sqrt\[(ac+bd)^2]
|z||w| = |ac + bd|

I don't know where the (ad-bc)^2 went or why the signs changed and changed back again...or why they're even doing it the way they are. Other explanations seem more longwinded but don't skip steps out. How would I 'prove' the triangle inequality in an exam btw?

The three things they take for granted are:

Can you see why these are so?

Hence, can you see how the jumped to the next step?
4. Here's an alternate way of thinking about it:

(ad-bc)^2 can only range from 0 to infinity.

Case 1: If (ad-bc)^2=0, then |z||w|=sqrt((ac+bd)^2).

Because these are the only two cases, we know that |z||w|≥sqrt((ac+bd)^2)

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Updated: April 6, 2013
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