Question

For any complex number z and w, show that
Iz - w| ≥ Izl - Iwl​

Answer 1

To show that for any complex numbers z and w, the inequality |z - w| ≥ |z| - |w| holds, we apply the triangle inequality for complex numbers and manipulate it to prove the desired result.

Step-by-step explanation:

To show that for any complex numbers z and w, the inequality |z - w| ≥ |z| - |w| holds, we will use the triangle inequality and some properties of complex numbers. The triangle inequality states that for any complex numbers z and w, the inequality |z + w| ≤ |z| + |w| is always true. By considering the complex number w as negative, we can rewrite this inequality as |z - w| ≥ |z| - |w|.

Let's start with the triangle inequality for complex numbers:

1. |z + (-w)| ≤ |z| + |-w|.
2. Since the magnitude of a complex number is always non-negative, we have |-w| = |w|.
3. Combining these two points, we obtain |z - w| ≤ |z| + |w|.
4. Since |-w| = |w|, we can rewrite the inequality as |z - w| ≥ |z| - |w|, which is what we are proving.

The triangle inequality holds for any complex numbers, and using this property, we are able to show the desired inequality.