Author Topic: Chapter 1.6 PG 63 ex Example 8  (Read 715 times)

Min Gyu Woo

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Chapter 1.6 PG 63 ex Example 8
« on: October 16, 2018, 10:17:54 PM »
Can someone explain how to use the triangle inequality to end up with

$$|z^2+4|\geq |z|^2-4$$

oighea

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Re: Chapter 1.6 PG 63 ex Example 8
« Reply #1 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.
« Last Edit: October 18, 2018, 03:35:42 AM by oighea »

Min Gyu Woo

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Re: Chapter 1.6 PG 63 ex Example 8
« Reply #2 on: October 17, 2018, 01:13:09 PM »
Triangle Inequality is

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