# What is the quotient of the complex number (4-3i)/(-1-4i)?

The aim of this question is to understand the simplification process of complex polynomials.

Such questions are solved by multiplying and dividing the given expression with the complex conjugate of the denominator.

The complex conjugate of a given expression say $( a \ + \ bi )$ is calculated simply by changing the sign of the imaginary part that is $( a \ – \ bi )$.

Given:

$\dfrac{ 4 \ – \ 3i }{ -1 \ – \ 4i }$

Multiplying and dividing by complex conjugate of $-1 \ – \ 4i$:

$\dfrac{ 4 \ – \ 3i }{ -1 \ – \ 4i } \times \dfrac{ -1 \ + \ 4i }{ -1 \ + \ 4i }$

$\Rightarrow \dfrac{ ( \ 4 \ – \ 3i \ )( \ -1 \ + \ 4i \ )}{ ( \ -1 \ – \ 4i \ )( \ -1 \ + \ 4i \ ) }$

$\Rightarrow \dfrac{ -4 \ + \ 3i \ + \ 16i \ – \ 12i^2 }{ ( \ -1 \ )^2 \ – \ ( \ 4i \ )^2 }$

$\Rightarrow \dfrac{ -4 \ + \ 19i \ – \ 12i^2 }{ 1 \ – \ 16i^2 }$

Substituting $i^2 \ = \ -1$:

$\dfrac{ -4 \ + \ 19i \ – \ 12 ( -1 ) }{ 1 \ – \ 16 ( -1 ) }$

$\Rightarrow \dfrac{ -4 \ + \ 19i \ + \ 12 }{ 1 \ + \ 16 }$

$\Rightarrow \dfrac{ 8 \ + \ 19i }{ 17 }$

$\Rightarrow \dfrac{ 8 }{ 17 } \ + \ \dfrac{ 19 }{ 17 } i$

## Numerical Result

$\dfrac{ 4 \ – \ 3i }{ -1 \ – \ 4i } \ = \ \dfrac{ 8 }{ 17 } \ + \ \dfrac{ 19 }{ 17 } i$

## Example

Find the quotient of the following complex number:

$\boldsymbol{ \dfrac{ 5 \ – \ 11i }{ 8 \ – \ 7i } }$

Multiplying and dividing by complex conjugate of $8 \ – \ 7i$:

$\dfrac{ 5 \ – \ 11i }{ 8 \ – \ 7i } \times \dfrac{ 8 \ + \ 7i }{ 8 \ + \ 7i }$

$\Rightarrow \dfrac{ ( \ 5 \ – \ 11i \ )( \ 8 \ + \ 7i \ )}{ ( \ 8 \ – \ 7i \ )( \ 8 \ + \ 7i \ ) }$

$\Rightarrow \dfrac{ 40 \ – \ 88i \ + \ 35i \ + \ 77i^2 }{ ( \ 8 \ )^2 \ – \ ( \ 7i \ )^2 }$

$\Rightarrow \dfrac{ 40 \ – \ 53i \ – \ 77i^2 }{ 64 \ – \ 49i^2 }$

Substituting $i^2 \ = \ -1$:

$\Rightarrow \dfrac{ 40 \ – \ 53i \ – \ 77 ( -1 )^2 }{ 64 \ – \ 49 ( -1 )^2 }$

$\Rightarrow \dfrac{ 40 \ – \ 53i \ + \ 77 }{ 64 \ + \ 49 }$

$\Rightarrow \dfrac{ 117 \ – \ 53i \ }{ 113 }$

$\Rightarrow \dfrac{ 117 }{ 113 } \ + \ \dfrac{ 53 }{ 113 } i$