Complex Modulus/Examples/z1 + z2

Example of Complex Modulus

Let $z_1 = 4 - 3 i$ and $z_2 = -1 + 2 i$.

Then:

$\cmod {z_1 + z_2} = \sqrt {10}$

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 $z_2$

\(\ds \)

\(=\)

\(\ds \cmod {\paren {4 - 1} + \paren {- 3 + 2} i}\)

Definition of Complex Addition

\(\ds \)

\(=\)

\(\ds \cmod {3 - i}\)

\(\ds \)

\(=\)

\(\ds \sqrt {3^2 + 1^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {10}\)

$\blacksquare$