Question

COMPLEX NUMBERS!

Answer 1

Prove algebraically that for complex numbers
\[|z_{1}| - |z_{2}| \le |z_{1} + z_{2} | \le |z_{1}| + |z_{2}| \]
Interpret this result in terms of two=dimentional vectors. Prove that:
\(|z-1| < |\sqrt{z^{2} - 1}|< |z+1|\) for R (z) > 0

Answer 2

have you idea, guys??

Answer 3

@myko

Answer 4

triangle inequality

Answer 5

here you go :
\(|z_1|=|z_1+z_2-z_2|\) now apply triangle inequality:
\(|z_1+z_2-z_2|\le |z_1+z_2|+|-z_2|=|z_1+z_2| +|z_2|\)
so: \(|z_1| -|z_2| \le |z_1+z_2|\)
this is for the left side. Right side is the triangle inequality

Answer 6

@gerryliyana

Answer 7

yeah.., i got it.., but how about second term.., ?

Answer 8

how about this :
\[|z-1| < |\sqrt{z^{2}-1} | < |z+1|\]
??

Answer 9

how about squaring each term and proving that otherwise is false.

Answer 10

so??