Skip to main content
Chemistry LibreTexts

2: Complex Numbers

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Chapter Objectives
    • Be able to perform basic arithmetic operations with complex numbers.
    • Understand the different forms used to express complex numbers (cartesian, polar and complex exponentials).
    • Calculate the complex conjugate and the modulus of a number expressed in the different forms (cartesian, polar and complex exponentials).
    • Be able to manipulate complex functions.
    • Be able to obtain expressions for the complex conjugate and the square of the modulus of a complex function.

    • 2.1: Algebra with Complex Numbers
      The imaginary unit i is defined as the square root of -1.
    • 2.2: Graphical Representation and Euler Relationship
      Complex numbers can be represented graphically as a point in a coordinate plane. In cartesian coordinates, the x -axis is used for the real part of the number, and the y -axis is used for the imaginary component. Complex numbers can be also represented in polar form.  We can also represent complex numbers in terms of complex exponentials.
    • 2.3: Complex Functions
      The concepts of complex conjugate and modulus that we discussed above can also be applied to complex functions.
    • 2.4: Problems

    Thumbnail: A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i satisfies \(i^2 = −1\). (CC BY-SA 3.0 unported; Wolfkeeper via Wikipedia)

    This page titled 2: Complex Numbers is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marcia Levitus via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.