6: Complex Numbers

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6.1: Complex Numbers
Although very powerful, the real numbers are inadequate to solve equations such as \(x^2+1=0\), and this is where complex numbers come in.
6.2: Polar Form
In the previous section, we identified a complex number \(z=a+bi\) with a point \(\left( a, b\right)\) in the coordinate plane. There is another form in which we can express the same number, called the polar form.
6.3: Roots of Complex Numbers
A fundamental identity is the formula of De Moivre with which we begin this section.
6.4: The Quadratic Formula
When working with real numbers, we cannot solve the quadratic formula if \(b^{2}-4ac<0.\) However, complex numbers allow us to find square roots of negative numbers, and the quadratic formula remains valid for finding roots of the corresponding quadratic equation.
6.E: Exercises

Thumbnail: Argument \(φ\) and modulus \(r\) locate a point in the complex plane. (CC BY-SA 3.0; Wolfkeeper via Wikipedia)