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Operations with Complex Numbers

To add two complex numbers , add the real part to the real part and the imaginary part to the imaginary part.

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

Example 1:

$\begin{array}{l} (2 + 7 i) + (3 - 4 i) = (2 + 3) + (7 + (- 4)) i \\ = 5 + 3 i \end{array}$

To subtract two complex numbers, subtract the real part from the real part and the imaginary part from the imaginary part.

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

Example 2:

$\begin{array}{l} (9 + 5 i) - (4 + 7 i) = (9 - 4) + (5 - 7) i \\ = 5 - 2 i \end{array}$

To multiply two complex numbers, use the FOIL method and combine like terms .

$\begin{array}{l} (a + b i) (c + d i) = a c + a d i + b c i + b d i^{2} \\ = a c + (a d + b c) i - b d (Remember i^{2} = - 1) \\ = (a c - b d) + (a d + b c) i \end{array}$

Example 3:

$\begin{array}{l} (3 + 2 i) (5 + 6 i) = 15 + 18 i + 10 i + 12 i^{2} \\ = 15 + 28 i - 12 \\ = 3 + 28 i \end{array}$

To divide two complex numbers, multiply the numerator and denominator by the complex conjugate , expand and simplify. Then, write the final answer in standard form.

$\frac{a + b i}{c + d i} \cdot \frac{c - d i}{c - d i} = \frac{(a c + b d) + (b c - a d) i}{c^{2} + d^{2}}$

Example 4:

$\begin{array}{l} \frac{3 + 2 i}{4 - 5 i} = \frac{3 + 2 i}{4 - 5 i} \cdot \frac{4 + 5 i}{4 + 5 i} \\ = \frac{12 + 15 i + 8 i + 10 i^{2}}{16 + 25} \\ = \frac{2 + 23 i}{41} \\ = \frac{2}{41} + \frac{23}{41} i \end{array}$