## Wednesday, December 14, 2005

### Studious Scholar - Complex Numbers

I'm currently completing a summer scholarship in applied mathematics. The project involves modelling ship-generated waves in two-dimensions using complex analysis. Say what? Let's start at the beginning.

We are all familiar with numbers from our everyday lives. You may have 2 apples, be 1.97 metres tall or have -21.58 dollars to your name. Normally, these regular or real numbers work fairly well. However, for mathematicians these numbers weren’t enough so they conceived a number such when multiplied by itself yields negative one. That is, a number, i, exists so i * i = -1. With the birth of this imaginary* number mathematicians suddenly had solutions to previously insolvable equations such as a*a = - 16 where a = 4i.

Mathematicians decided to combine our familiar real numbers with these new imaginary numbers to obtain complex numbers such as 4+2i or -10+16i. Complex numbers, such as 3+4i, have a real part and an imaginary part (in our example 3 and 4 respectively). Amazingly, complex numbers are more than just a mathematical curiosity. Their applications are wide and varied. In fact, complex numbers are used to model electronic circuits, predict atomic and chemical reactions, and even explain the mysteries of the cosmos.

In addition to these applications, when you combine complex numbers with old high school calculus (an algebraic approach to geometry), you obtain the very elegant field known as complex analysis – a powerful branch of mathematics that behaves differently to regular calculus.

And it just happens that complex number-based equations behave exactly like the equations used to model the flow of heat, electricity and fluid in two dimensions – where the real part (of the complex number) represents one dimension, say width, and the imaginary part represents the other dimension, say height.

Since mathematicians have studied complex numbers and their equations rigorously for the last couple centuries, there is a wide range of handy tools available for solving equations that describe physical systems. Thus, mathematicians can solve fluid mechanics 2-D problems (admittedly, our spatial universe is 3-D but suprisingly, often 2-D models are sufficient).

My summer project involves deriving and solving these complex number based equations, and subsequently simulating on a computer the flow of water in an attempt to gain some insight on how to design a more efficient ship stern.

*Imaginary and real are poor terms as students tend to think that real numbers exist where imaginary numbers are purely inventions by mathematicians. In fact both real and imaginary numbers are inventions by mathematicians.