Divide the following complex numbers.
\qquad \dfrac{A_REP}{B_REP}
Since we're dividing by a single term, we can simply divide each term in the numerator separately.
\qquad
\dfrac{A_REP}{B_REP} = \dfrac{A_REAL}{B_REP}
SIGN \dfrac{coefficient(abs(A_IMAG))i}{B_REP}
Factor out \dfrac 1i.
\qquad = \dfrac 1i \left( \dfrac{A_REAL}{B_IMAG} SIGN
\dfrac{coefficient(abs(A_IMAG))i}{B_IMAG} \right)
\qquad = \dfrac 1i (complexNumber(-ANSWER_IMAG, ANSWER_REAL))
\qquad \dfrac 1i = \dfrac 1i \cdot \dfrac ii = \dfrac{1 \cdot i}{i \cdot i} = \dfrac{i}{-1} = -i
Substitute -i for \dfrac 1i:
\qquad \begin{eqnarray}
&=& -i (complexNumber(-ANSWER_IMAG, ANSWER_REAL)) \\
&=& ANSWER_IMAGi + -ANSWER_REALi^2 \\
&=& ANSWER_REP
\end{eqnarray}
We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate, which is \green{CONJUGATE}.
\qquad \dfrac{A_REP}{B_REP} =
\dfrac{A_REP}{B_REP} \cdot
\dfrac{\green{CONJUGATE}}{\green{CONJUGATE}}
We can simplify the denominator using the fact (a + b) \cdot (a - b) = a^2 - b^2.
\qquad = \dfrac{(A_REP) \cdot (CONJUGATE)}
{negParens(B_REAL)^2 - (coefficient(B_IMAG)i)^2}
Evaluate the squares in the denominator and subtract them.
\qquad = \dfrac{(A_REP) \cdot (CONJUGATE)}
{(B_REAL)^2 - (coefficient(B_IMAG)i)^2}
\qquad = \dfrac{(A_REP) \cdot (CONJUGATE)}
{B_REAL * B_REAL + B_IMAG * B_IMAG}
\qquad = \dfrac{(A_REP) \cdot (CONJUGATE)}
{B_REAL * B_REAL + B_IMAG * B_IMAG}
The denominator now doesn't contain any imaginary unit multiples, so it is a real number.
Note that when a complex number, a + bi is multiplied by its conjugate,
the product is always a^2 + b^2.
Now, we can multiply out the two factors in the numerator.
\qquad \dfrac{(\blue{A_REP}) \cdot (\red{CONJUGATE})}
{DENOMINATOR}
\qquad = \dfrac{\blue{A_REAL} \cdot \red{negParens(B_REAL)} + \blue{A_IMAG} \cdot \red{negParens(B_REAL) i} + \blue{A_REAL} \cdot \red{B_CONJUGATE_IMAG i} + \blue{A_IMAG} \cdot \red{B_CONJUGATE_IMAG i^2}}
{DENOMINATOR}
\qquad = \dfrac{A_REAL * B_REAL + A_IMAG * B_REALi + A_REAL * B_CONJUGATE_IMAGi + A_IMAG * B_CONJUGATE_IMAG i^2}
{DENOMINATOR}
Finally, simplify the fraction.
\qquad \dfrac{A_REAL * B_REAL + A_IMAG * B_REALi + A_REAL * B_CONJUGATE_IMAGi - A_IMAG * B_CONJUGATE_IMAG}
{DENOMINATOR} =
\dfrac{REAL_NUMERATOR + IMAG_NUMERATORi}
{DENOMINATOR} =
ANSWER_REP