Toronto Math Forum
MAT3342018F => MAT334Lectures & Home Assignments => Topic started 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
$$z^2+4\geq z^24$$

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.

Triangle Inequality is
$$z+w \leq z + w$$