Preface

Formal Definition: A complex number is a number comprising a real and imaginary part (${\displaystyle z=x+yi}$), where x and y are real numbers, and i is the standard imaginary unit with the property ${\displaystyle i^{2}=-1,i=\pm {\sqrt {-1}}}$, a property that has led to the discovery of the fundamental theorem of algebra. The complex number field (${\displaystyle \mathbb {R} \in \mathbb {C} }$) is a superset of the real number field.

Translation: i represents numbers that don't exist, and acts as a convenient placeholder, thus allowing one to get the roots of any polynomial. The fundamental theorem of algebra states that any polynomial as the same number of roots, as its highest power. All complex numbers are examples of vectors on a different plane, commonly called an Argand diagram.