2. Basic Operations of Complex Number

For $i^2=-1$ and any real numbers $a$, $b$, $c$ and $d$.

1. Addition of complex number

   $(a+ib)+(c+id)=(a+c)+i(b+d)$

   Simplify $(4+5i)+(8-2i)$.

   $\begin{array}{cl}
   & (4+5i)+(8-2i) \\
   = & (4+8)+i(5+(-2)) \\
   = & 12+3i
   \end{array}$
2. Subtraction of complex number

   $(a+ib)-(c+id)=(a-c)+i(b-d)$

   Simplify $(3-2i)-(2+7i)$.

   $\begin{array}{cl}
   & (3-2i)-(2+7i) \\
   = & (3-2)+i((-2)-7) \\
   = & 1-9i
   \end{array}$

3. Multiplication of complex number

   $(a+ib)(c+id)=(ac-bd)+i(bc+ad)$

   Simplify $(5-2i)(3+9i)$.

   $\begin{array}{cl}
   & (5-2i)(3+9i) \\
   = & (5)(3)+(5)(9i)+(-2i)(3)+(-2i)(9i) \\
   = & 15+45i-6i-18i^2 \\
   = & 15+39i-18(-1) \\
   = & 33+39i
   \end{array}$

4. Division of complex number

   $\dfrac{a+ib}{c+id}=\dfrac{(ac+bd)+i(bc-ad)}{c^2+d^2}$

   Simplify $\dfrac{2-5i}{2+7i}$.

   $\begin{array}{cl}
   & \dfrac{2-5i}{2+7i} \\
   = & \dfrac{2-5i}{2+7i} \times \dfrac{2-7i}{2-7i} \\
   = & \dfrac{(2-5i)(2-7i)}{(2+7i)(2-7i)} \\
   = & \dfrac{(2)(2)+(2)(-7i)+(-5i)(2)+(-5i)(-7i)}{2^2-(7i)^2} \\
   = & \dfrac{4-14i-10i+35i^2}{4-49i^2} \\
   = & \dfrac{4-24i+35(-1)}{4-49(-1)} \\
   = & \dfrac{-31-24i}{53} \\
   = & \dfrac{-31}{53}-\dfrac{24}{53}i
   \end{array}$

Same Topic:

1. Definitions Discriminant Inverse Variation Square Numbers