Complex Numbers

Overview

Engineering reference data for Complex Numbers in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Operations on Complex Numbers

Addition

In Cartesian form, complex numbers are added by separately summing their real and imaginary parts: $Z_a + Z_b = (a + c) + j(b + d)$

In polar form, addition is performed by: $Z_a + Z_b = (r_a \cos\theta_a + r_b \cos\theta_b) + j(r_a \sin\theta_a + r_b \sin\theta_b)$

Example:
$Z_a = 3 + j2$, $Z_b = -3 + j3$ → $Z_a + Z_b = j5$

Subtraction

In Cartesian form: $Z_a - Z_b = (a - c) + j(b - d)$

Example:
$Z_a = 3(\cos35° + j\sin35°)$, $Z_b = 2(\cos15° + j\sin15°)$ → $Z_a - Z_b = 0.52 + j1.2$

Multiplication

In Cartesian form: $Z_a \cdot Z_b = (ac - bd) + j(ad + bc)$

In polar form, multiplication is simplified using modulus and argument: $Z_a \cdot Z_b = r_a r_b \left[ \cos(\theta_a + \theta_b) + j \sin(\theta_a + \theta_b) \right]$

Example:
$Z_a = 3 + j2$, $Z_b = 5 - j4$ → $Z_a \cdot Z_b = 23 - j2$

Complex Conjugate

The complex conjugate of $Z = a + jb$ is $Z^* = a - jb$. Multiplying a complex number by its conjugate yields a real number: $Z \cdot Z^* = a^2 + b^2$

Example:
$Z = 3 + j2$, $Z^* = 3 - j2$ → $Z \cdot Z^* = 13$

Division

Division is performed by rationalizing the denominator using the conjugate: $\frac{Z_a}{Z_b} = \frac{ac + bd}{c^2 + d^2} + j\frac{bc - ad}{c^2 + d^2}$

In polar form, division simplifies to: $\frac{Z_a}{Z_b} = \frac{r_a}{r_b} \left[ \cos(\theta_a - \theta_b) + j \sin(\theta_a - \theta_b) \right]$

References

• Original Source