B Complex Numbers


The following videos explain why imaginary numbers are necessary in mathematics. Note that these videos use \(i\) to denote the imaginary number \(\sqrt{-1}\), whereas we use \(j\) (which is common in electrical engineering, to avoid confusion with current).


The following videos explain the core concepts: the complex plane and the geometric interpretations of complex addition and multiplication.

Be sure to try the calculations suggested at the end of the previous video before moving on to the next video:

  • \((4 + 3j) \cdot j\)
  • \((4 + 3j) \cdot 2j\)
  • \((4 + 3j) \cdot (4 + 3j)\)
  • \((2 + j) \cdot (1 + 2j)\)
Definition B.1 (Magnitude of a Complex Number) The magnitude of a complex number \(z = a + jb\) is \[ |z| \overset{\text{def}}{=} \sqrt{a^2 + b^2}. \] This follows from the Pythagorean Theorem.
Definition B.2 (Complex Conjugate) The complex conjugate of a complex number \(z = a + jb\) is \[ z^* \overset{\text{def}}{=} a - jb. \]

To conjugate a complex number, we simply flip the sign in front of any \(j\)s in the expression. For example: \[ \left(\frac{1 + 2j}{2 - j} \right)^* = \frac{1 - 2j}{2 + j}. \]

Because it is so easy to calculate the complex conjugate, conjugation is often the preferred way to calculate magnitudes.

Theorem B.1 (Calculating Magnitude) The squared magnitude of a complex number \(z\) is the product of the number and its complex conjugate: \[ |z|^2 = z z^*. \] This means that the magnitude of a complex number can be calculated as \[ |z| = \sqrt{z z^*}. \]
Proof. If \(z = a + jb\), then \(z^* = a - jb\), and \[ z z^* = (a + jb)(a - jb) = a^2 - j^2 b^2 = a^2 + b^2 = |z|^2. \]

For example, the magnitude of the number above is: \[ \left|\frac{1 + 2j}{2 - j} \right| = \sqrt{\frac{1 + 2j}{2 - j} \cdot \frac{1 - 2j}{2 + j}} = \sqrt{\frac{1 + 4}{4 + 1}} = 1. \] It would have been much more tedious to calculate this magnitude by finding \(a\) and \(b\) such that \(\frac{1 + 2j}{2 - j} = a + jb\) and using Definition B.1.

Theorem B.2 (Euler’s Identity) Euler’s identity relates the complex exponential \(e^{j\theta}\) to the trigonometric functions: \[ e^{j\theta} = \cos \theta + j \sin\theta \] Two immediate corollaries of Euler’s identity are: \[\begin{align*} \cos\theta &= \frac{e^{j\theta} + e^{-j\theta}}{2} \\ \sin\theta &= \frac{e^{j\theta} - e^{-j\theta}}{2j}. \end{align*}\]

Essential Practice

  1. Express the complex number \(\frac{1 - j}{1 + j}\) in \(a + jb\) form, where \(a\) and \(b\) are real numbers.
  2. Calculate \(e^{j\pi/4} + e^{-j\pi/4}\). Your answer should be a real number.
  3. Calculate \(\left| \frac{1 - j}{1 + e^{j\pi/4}} \right|\).