Answer
The division of a complex number requires the multiplication of the complex conjugate of the denominator in the numerator and denominator, and then we use the FOIL method to simplify.
Work Step by Step
Procedure to divide the two complex numbers:
1. Multiply the numerator and the denominator by the complex conjugate of the denominator.
2. Use the FOIL method in the numerator and the denominator.
3. Replace the value ${{i}^{2}}=-1$.
4. Combine the real terms and combine the imaginary terms.
5. Express the answer in standard form.
FOIL method to multiply the four terms of the product.
\[\left( a+b \right)\left( c+d \right)=\overbrace{ac}^{\text{F}}+\overbrace{ad}^{\text{O}}+\overbrace{bc}^{\text{I}}+\overbrace{bd}^{\text{L}}\]
F is the first terms of each binomial.
O is the outside terms or first term of the first binomial and second term of the second binomial.
I is the inside terms or second term of the first binomial and first term of the second binomial.
L is the last terms of each binomial.
The standard form of a complex number is $a+bi$, where $a$ is the real part and $b$ is the imaginary part.
For example,
Consider the complex numbers, $\left( 2+3i \right)$ and $\left( 2+i \right)$
Divide the complex numbers as,
$\frac{2+3i}{2+i}$
Multiply by complex conjugate of the denominator in the numerator and the denominator.
$\frac{2+3i}{2+i}=\frac{\left( 2+3i \right)}{\left( 2+i \right)}\cdot \frac{\left( 2-i \right)}{\left( 2-i \right)}$
Use the FOIL method.
\[\begin{align}
& \frac{2+3i}{2+i}=\frac{\left( 2+3i \right)\left( 2-i \right)}{\left( 2+i \right)\left( 2-i \right)} \\
& =\frac{4-2i+6i-3{{i}^{2}}}{4-2i+2i-{{i}^{2}}} \\
& =\frac{4+4i-3{{i}^{2}}}{4-{{i}^{2}}}
\end{align}\]
Replace the value ${{i}^{2}}=-1$.
\[\begin{align}
& \frac{2+3i}{2+i}=\frac{4+4i-3\left( -1 \right)}{4-\left( -1 \right)} \\
& =\frac{4+4i+3}{4+1}
\end{align}\]
Make a group of real and imaginary terms.
\[\begin{align}
& \frac{2+3i}{2+i}=\frac{\left( 4+3 \right)+4i}{5} \\
& =\frac{7+4i}{5}
\end{align}\]
Express the complex number in the standard form.
\[\frac{2+3i}{2+i}=\frac{7}{5}+\frac{4}{5}i\]
Therefore, the division of a complex number requires the multiplication of the complex conjugate of the denominator in the numerator and denominator, and then we use the FOIL method to simplify.
