Complex Modulus/Examples/z1 - z2
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Example of Complex Modulus
Let $z_1 = 4 - 3 i$ and $z_2 = -1 + 2 i$.
Then:
- $\cmod {z_1 - z_2} = 5 \sqrt 2$
Proof 1
An illustration of the modulus of the sum of the complex numbers:
- $z_1 = 4 - 3 i$
- $z_2 = -1 + 2 i$
is given below:
The modulus is seen to be the radius of the circle.
$\blacksquare$
Proof 2
\(\ds \cmod {z_1 - z_2}\) | \(=\) | \(\ds \cmod {\paren {4 - 3 i} - \paren {-1 + 2 i} }\) | Definition of $z_1$ and $z_2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\paren {4 - \paren {-1} } + \paren {-3 - 2} i}\) | Definition of Complex Subtraction | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {5 - 5 i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {5^2 + 5^2}\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(=\) | \(\ds 5 \sqrt 2\) |
$\blacksquare$