Complex Arithmetic/Examples/Modulus of ((z 1 + z 2 + 1) (z 1 - z 2 + i)^-1)

Example of Complex Arithmetic

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

Then:

$\cmod {\dfrac {z_1 + z_2 + 1} {z_1 - z_2 + i} } = \dfrac 3 5$

$\ds \cmod {\dfrac {z_1 + z_2 + 1} {z_1 - z_2 + i} }$

$=$

$\ds \cmod {\dfrac {\paren {1 - i} + \paren {-2 + 4 i} + 1} {\paren {1 - i} - \paren {-2 + 4 i} + i} }$

$\ds $

$=$

$\ds \cmod {\dfrac {\paren {3 i} } {\paren {3 + 4 i} } }$

$\ds $

$=$

$\ds \dfrac {\cmod {3 i} } {\cmod {3 + 4 i} }$

$\ds $

$=$

$\ds \dfrac 3 {\sqrt {3^2 + 4^2} }$

$\ds $

$=$

$\ds \dfrac 3 5$

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Fundamental Operations with Complex Numbers: $54 \ \text {(e)}$