• 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 \(\textbf{A}-\textbf{B}=(a – c) + j(b – d)\)
    Multiplication \(\textbf{A}\textbf{B}=(ac – bd) + j(bc + ad)\)
    Conjugate \(\textbf{A*}=a–jb\)
    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
    Addition Convert to rectangular
    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
    Addition Convert to rectangular
    Subtraction Convert to rectangular
    Multiplication \(\textbf{A}\textbf{B}=XY\angle(\theta+\phi)\)
    Division \(\textbf{A}/\textbf{B}=\frac{X}{Y}\angle(\theta-\phi)\)




    Last updated:
    January 6, 2018