Prove that: #|z_1+z_2+z_3+.......................+z_n|=|z_1|+|z_2|+|z_3|+...............+|z_n|?#
2 Answers
I don't think that equation is valid. I'm assuming
Explanation:
Try with two terms,
Hence
Perhaps you mean the triangle inequality for complex numbers:
We can abbreviate this
where the sums are
Lemma.
The real part is never bigger than the magnitude. Let
I'll use the overbar for conjugate. Here we have a real number, the squared magnitude, which equals the product of the conjugates. The trick is that it equals its own real part. The real part of the sum is the sum of the real parts.
By our lemma, and the magnitude of the product being the product of magnitudes, and the magnitude of conjugates are equal,
We can cancel one factor of the magnitude of the sum
That's what we wanted to prove.