Chapter: Vector Properties of Complex Numbers

A useful way to deal with complex numbers numbers providing many new insights is to regard them as a vector space. If you are already acquainted with vectors, you will probably have noticed the parallel from complex numbers to vectors when looking at the way we add complex numbers. In this chapter, we will explore this topic more systematically. We will also introduce the absolute value of complex numbers, which can be regarded as the length of a complex number vector.

  1. Lemma: Linear Independence of the Imaginary Unit \(i\) and the Complex Number \(1\)
  2. Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers \(1\) and Imaginary Unit \(i\) Form Their Basis
  3. Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers
  4. Definition: Complex Conjugate
  5. Definition: Dot Product of Complex Numbers
  6. Definition: Absolute Value of Complex Numbers
  7. Definition: Argument of a Complex Number
  8. Proposition: Polar Coordinates of a Complex Number
  9. Proposition: Multiplication of Complex Numbers Using Polar Coordinates

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