Helpful Tools for Calculus, Chapter 0: Numbers, Notations and Notes
Previous Section: 0.1 Real Numbers
Although a majority of Calculus will focus on the Real Numbers, it is important and interesting to understand the historical development and usage of Complex Numbers. This section will provide a background to the creation and understanding of the numbers we call imaginary.
16th Century
In 1545 Gerolamo Cardano introduced and dismissed complex numbers as useless. He disliked them and compared them to torture. In evaluating some cubic equations he identified some that generally had one real number solution and two complex number solutions. Like previous mathematicians, he regarded these as useless and ignored them. However, he spurred on work about the general polynomials which led to the Fundamental Theorem of Algebra which states that a solution exists to every polynomial equation of degree one or higher, including complex solutions.
17th Century
These numbers weren’t referred to as imaginary until 1637 when René Descartes who was attempting to understand what a complex solution would look like and represent. With no notation for 
numerous algebraic mistakes began to occur, like the following:
1=√1 √1
=√(12 )
=√[(-1)2]
=√(-1) √(-1)
= -1
This is wrong because the formula √ab= √a √b only applies for a,b≥0.
18th Century
During the 1700s many breakthroughs were made in what would later be called complex analysis. Many mathematicians noticed that manipulation of complex expressions could be used to simplify and rewrite other types of functions like those in trigonometry. However, some notation needed to occur first.
In order to guard against the error above, Leonhard Euler invented the notation i= and he believed that complex numbers should be introduced to students much earlier than we do today. He introduced the complex numbers early in his elementary algebra textbook and used them throughout in a coherent and natural way.
In 1730 Abraham de Moivre shows 
which is known de Moivre’s Formula. Then in 1748 Euler stated 
which is known as Euler’s Formula of Complex Analysis. This is often referred to as the most beautiful equation in mathematics.
This century also introduced the idea of using a complex plane to describe these numbers. Almost like a third dimension, we could plot points in the complex realm by adding an axis that refers to the value multiplied by i in an equation (2+3i for example).
19th Century and on
Carl Friedrich Gauss is often credited with being the first to use complex numbers scientifically and confidently. This ushered in a new wave of mathematicians and scientists using these imaginary numbers in more interesting and applicable ways and bringing complex analysis to an understandable state. Some of these mathematicians even expanded the complex number system to represent new types of complex numbers (such as i, j, k) to refer to a quaternion number system. The Hamiltonian mechanics represented a quotient of two vectors and can be used for three-dimensional rotations, particularly in computer graphics!
Review
Mathematicians have been studying Complex Numbers since the 16th century and have recently been expanded to create several different definitions for . Most often we will call this i so that complex numbers are represented by a+bi where a,b∈
.