With further teachings about the complex numbers, the lecture keeps elaborating on solving certain differential equations by using complex numbers. After some of my own research, I found the history of complex numbers quite interesting.

It all started algebraically. Quadratic equations are one of the most common equations in math. It was used and studies for thousands of years. It usually appears in the form of ax^{2}+bx+c=0, and naturally the formula for x would appear as following,.

Then people realized that sometimes the equation has no solution, or has no real solution, as we may say today. In fact, there should always be two solutions for quadratic equations, it’s just sometimes they are not real. By the time of 1637, the famous mathematician, Rene Descartes, invented the concept of “imaginary number”, the square root of -1, as I mentioned in the previous blog. Personally speaking, I think the most possible reason for the invention of complex number is the justification of all kinds of unsolvable functions. Again, for the quadratic equations with 4ac>b^{2}, no real answer can be reached, but quadratic equations are such basic and important equation in science world that without the solution of which no further calculation can be done.

When the unsolvable nature of certain equations become an obstacle for the progress of science, something had to be done, and complex numbers were invented to fill in the blank so that the calculation could go on.

With the development of mathematics, more about the nature of complex numbers was discovered. Gradually, mathematicians tried to associate the algebraic concept with geometric ones. In 1748, Euler discovered what we call “Euler’s formula” today: e^{ix}=conx+i*sinx, which was introduced in the lecture. The proving process, although not so hard to understand nowadays, was highly sophisticated then (for the specifics about the proving process of Euler’s formula, click here). Nonetheless, Euler’s formula was a huge leap forward, because with such interpretation, any exponential functions with complex number involved can be solved easily.

About another 100 years passed, people started seeking for a geometric interpretation of complex numbers. It is hard to visualize something as highly abstract as the idea of complex numbers. The Norwegian mathematician Wessel found it first. He did so by imagining every complex number as a dot on a coordinate system. In this coordinate system, the horizontal axis is the “real” axis and the vertical axis is the “imaginary” axis. When a complex number is expressed as “a+bi”, it can be easily located on the axis. Interestingly, the expression for complex numbers is two-dimensional, unlike the one-dimensional nature of that for real numbers. What’s more interesting about such interpretation of complex number is the polar expression of “a+bi”. The angle “q” between the line from “a+bi” to the origin and the real axis and the distance of that line “r” are used in the expression: a+bi=r(cosq+i*sinq), which looks extremely like the Euler’s formula. (for more information of the geometric interpretation of complex numbers, click here)

picture from http://mathwiki.ucdavis.edu/Algebra/Linear_algebra/02._Introduction_to_complex_numbers/2.3_Polar_Form_and_Geometric_Interpretation

Although I understand Euler’s formula, I am still confused about the complex method of solving differential equations. I will keep studying the topic until my questions are answered.