• BME 210
• # Handout 6: Complex Numbers

There are three ways to represent complex numbers: rectangular, exponential, and polar or phasor. Addition and subtraction are easier with rectangular while multiplication and division are easier with exponential and polar.

## Rectangular Form

Given $$\textbf{A}=a+jb$$ and $$\textbf{B}=c+jd$$

Operation
Addition $$\textbf{A}+\textbf{B}=(a + c) + j(b + d)$$
Subtraction $$\textbf{A}-\textbf{B}=(a â c) + j(b â d)$$
Multiplication $$\textbf{A}\textbf{B}=(ac â bd) + j(bc + ad)$$
Conjugate $$\textbf{A*}=aâjb$$
Division $$\frac{\textbf{A}}{\textbf{B}}=\frac{\textbf{A}\textbf{B*}}{\textbf{B}\textbf{B*}}=\frac{(ac+bd) + j(bc-ad)}{c^2+d^2}$$

## Exponential Form

Given $$\textbf{A}=a+jb$$ and $$\textbf{B}=c+jd$$

$$X=\sqrt{a^2+b^2}$$ and $$\theta=\tan^{-1}\frac{b}{a}$$

$$Y=\sqrt{c^2+d^2}$$ and $$\phi=\tan^{-1}\frac{d}{c}$$

$$\textbf{A}=Xe^{j\theta}$$ and $$\textbf{B}=Ye^{j\phi}$$

Operation
Subtraction Convert to rectangular
Multiplication $$\textbf{A}\textbf{B}=XYe^{j(\theta+\phi)}$$
Division $$\textbf{A}/\textbf{B}=\frac{X}{Y}e^{j(\theta-\phi)}$$

## Polar Form

Given $$\textbf{A}=a+jb$$ and $$\textbf{B}=c+jd$$

$$\textbf{A}=X\angle\theta$$ and $$\textbf{B}=Y\angle\phi$$

Operation
Multiplication $$\textbf{A}\textbf{B}=XY\angle(\theta+\phi)$$
Division $$\textbf{A}/\textbf{B}=\frac{X}{Y}\angle(\theta-\phi)$$