Throughout this chapter, we gained an understanding of complex numbers, #z=a+b\cdot \ii#, and their fundamental operations. These included the usual operations of addition, subtraction, multiplication, and division, as well as new operations like complex conjugation, #\bar{z}=a-b\cdot \ii#.

We also learned how to represent the real and imaginary parts of complex numbers, both graphically and analytically. Doing so, we learned some operations are easier when complex numbers are in cartesian form, while others are better suited for numbers written in polar form. We write a number #z# in polar form,
\[\begin{array}{rcl} z&=&r\cdot\left(\cos\left(\theta\right)+\sin\left(\theta\right)\cdot \ii\right)\\
a&=&r\cdot\cos\left(\theta\right)\\
b&=&r\cdot\sin\left(\theta\right)\end{array}\]

where #r# is the norm of #z# and #\theta# is the angle of #z#, measured counter-clockwise from the positive #x#-axis.

This knowledge allows us to explore more advanced topics where complex numbers are vital. These include studying complex-valued functions like exponentials, logarithms, and polynomials. In addition to functions, we can also study Euler’s representation, which connects the polar form with complex exponentials, and De Moivre’s theorem, which simplifies the computation of complex roots. This enriches our understanding of cyclic phenomena and polynomial equations in mathematical and scientific contexts.