Triangle Inequality/Complex Numbers/Examples/3 Arguments/Proof 2
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Example of Use of Triangle Inequality for Complex Numbers
For all $z_1, z_2, z_3 \in \C$:
- $\cmod {z_1 + z_2 + z_3} \le \cmod {z_1} + \cmod {z_2} + \cmod {z_3}$
Proof
| \(\ds \cmod {z_1 + z_2 + z_3}\) | \(=\) | \(\ds \cmod {z_1 + \paren {z_2 + z_3} }\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \cmod {z_1} + \cmod {z_2 + z_3}\) | Triangle Inequality for Complex Numbers | |||||||||||
| \(\ds \) | \(\le\) | \(\ds \cmod {z_1} + \cmod {z_2} + \cmod {z_3}\) | Triangle Inequality for Complex Numbers |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $7 \ \text{(b)}$