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Chemistry LibreTexts

2: Complex Numbers

  • Page ID
    106811
  • 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)

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