# Toronto Math Forum

## MAT334-2018F => MAT334--Lectures & Home Assignments => Topic started by: Min Gyu Woo on October 16, 2018, 10:17:54 PM

Title: Chapter 1.6 PG 63 ex Example 8
Post by: Min Gyu Woo on October 16, 2018, 10:17:54 PM
Can someone explain how to use the triangle inequality to end up with

$$|z^2+4|\geq |z|^2-4$$
Title: Re: Chapter 1.6 PG 63 ex Example 8
Post by: oighea on October 16, 2018, 10:44:40 PM
Let $a = z^2, b = 4$. By the triangle inequality, $|a + b| \leq |a| + |b|$. Therefore, by substitution, $|z^2 + 4| < |z^2| + 4 < |z^2| - 4$, and note $|z^2| == |z|^2$.

To visualize the triangle inequality in the complex plane, the lengths of the two sides of the origin are given as $|a|, |b|$, and the third side is $|a + b|$. As expected, the third side is shorter than the sum of the two sides.

The complex triangle inequality can be proved by squaring.
Title: Re: Chapter 1.6 PG 63 ex Example 8
Post by: Min Gyu Woo on October 17, 2018, 01:13:09 PM
Triangle Inequality is

$$|z+w| \leq |z| + |w|$$