Complex Arithmetic/Examples/1 over 1+i Plus 1 over 1-2i

Example of Complex Arithmetic

$\dfrac 1 {1 + i} + \dfrac 1 {1 - 2 i} = \dfrac 7 {10} - \dfrac 1 {10} i$

Proof

\(\ds \dfrac 1 {1 + i} + \dfrac 1 {1 - 2 i}\)

\(=\)

\(\ds \dfrac {1 - i} {\left({1 + i}\right) \left({1 - i}\right)} + \dfrac {1 + 2 i} {\left({1 - 2 i}\right) \left({1 + 2 i}\right)}\)

multiplying top and bottom by conjugate of bottom

\(\ds \)

\(=\)

\(\ds \dfrac {1 - i} {1^2 + 1^2} + \dfrac {1 + 2 i} {1^2 + 2^2}\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {1 - i} 2 + \dfrac {1 + 2 i} 5\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {5 \left({1 - i}\right) + 2 \left({1 + 2 i}\right)} {10}\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {5 - 5 i + 2 + 4 i} {10}\)

simplifying

\(\ds \)

\(=\)

\(\ds \dfrac {7 - i} {10}\)

simplifying

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: Example $2$.