# Complex Numbers

A complex number can be represented graphically in the complex plane.

This point can be described in several different ways: $x+jy = r e^{\j \theta} = r cos\theta + r j sin\theta$

Note that the third relationship is derived when substituting these trigonometric functions into the first relationship: $x = r cos\theta \quad \& \quad y = r sin\theta$

Also: $(x ^ {2} + y ^{2}) ^ { \frac{1}{2}} = r \quad \& \quad \theta = tan ^ {-1}( \frac{y}{x})$

Rectangular: $z_{1} = x_{1} + j y_{1} \quad \& \quad z_{2} = x_{2} + jy_{2}$ $z_{1} + z_{2} = (x_{1} + x_{2}) + j( y_{1} + y_{2})$ $z_{1} - z_{2} = (x_{1} - x_{2}) + j( y_{1} - y_{2})$

Polar: There is no simple way of doing this. The best method would be to convert to rectangular form using the formula: $r cos\theta + r j sin\theta$

### Multiplication/Division of Complex Numbers:

Rectangular: $z_{1} = x_{1} + j y_{1} \quad \& \quad z_{2} = x_{2} + jy_{2}$ $z_{1} \cdot z_{2} = (x_{1} + j y_{1})(x_{2} + j y_{2}) = x_{1}x_{2} + j(x_{2}y_{1} + x_{1}y_{2}) - y_{1}y_{2}$ $\frac{z_{1}}{z_{2}} = \frac{x_{1} + j y_{1}}{x_{2} + jy_{2}}(\frac{x_{2} - j y_{2}}{x_{2} - jy_{2}}) = \frac{x_{1}x_{2} + j(x_{2}y_{1} - x_{1}y_{2}) + y_{1}y_{2}}{x_{2}^{2}+y_{2}^{2}}$

When multiplying two complex numbers in rectangular form, the FOIL method must be used. When dividing complex numbers in rectangular form, multiply the numerator and denominator by the complex conjugate of the denominator.

Polar: $z_{1} = c_{1} e ^ {j \theta_{1}} \quad \& \quad z_{2} = c_{2} e ^ {j \theta_{2}}$ $z_{1} \cdot z_{2} = (c_{1} e ^ {j \theta_{1}})(c_{2} e ^ {j \theta_{2}}) = c_{1} c_{2} e ^ {j(\theta_{1} + \theta_{2})}$ $\frac{z_{1}}{z_{2}} = (\frac{c_{1}}{c_{2}}) e ^ {j(\theta_{1} - \theta_{2})}$