Expert: Josh - 5/20/2009

Question
Hi Josh,

Why is it important to find the roots of equations..in a practical application.Same for complex numbers..what is so special abt complex numbers and how is it applied in real life?Thanks

Answer
Hi Yee,

This is an excellent question. Let's look at this in two parts. First, the practical significance of solving an equation. Second, the ability to find the roots of an equation.

First, we can look at equations as a by-product of mathematical modelling. "Modeling" is the process of capturing important characteristics of a complex problem and transforming it into a simpler form. Mathematical equations provide an elegant way of representing some underlying problem, capturing some aspect(s) of interest and expressing it in a "universal" language that can be understood by many. Another reason for converting a problem into a mathematical formulation is that it gives us greater insight into a problem, and the tools for finding the answer to specific questions.

This has virtually unlimited potential, because the problems and questions can take many forms. Modelling and analysis can be applied in many practical domains (from physics, engineering to finance etc.)

In engineering, people are generally interested in finding an optimal way of using a limited pool of resources to achieve the best outcome. Consider a construction project for example. The variables (unknown quantities in the equations) may represent the amount of each material to use and there is a cost associated with each material. The goal might be to find the best combination that
- minimises a) the total cost, b) structural risk, or
- maximises a) energy efficiency, b) strength of a building...
and so forth

This type of enquiry is referred to as "optimization". In a totally different sphere, such as finance, a person may set up a set of (not-so-simple, non-linear) equations to represent his investment portfolio and options. Solving the equations in this instance amount to searching for the optimal parameters which maximize the return on investment. Here, the "x,y,z etc" each relates to some decision; or perhaps, how the money should be divided between different classes of investment.

Further Reading: Calculus is a branch of mathematics that deals with the standard problem of "maximising" (or minimising) something. In high school, you learn about methods of "differentiation". This gives you the grounding for learning more advanced techniques later at university (if you study engineering, science, economics etc).

The second part of the puzzle is whether it is always possible to find solutions given a polynomial equation. Informally, the fundamental theorem of algebra tells us that this is possible, because a polynomial of degree n always has n roots (although some may be complex numbers). This, in part, explains why complex numbers are so special.

There are too much examples to cite regarding the practical application of complex numbers. The best way to look at it, is again, from the perspective of "modelling and analysis" introduced earlier. An important class of technique called Fourier analysis makes extensive use of complex numbers. This is used in many different fields, and arguably, one of the most important application is in the transmission and representation of digital information. It is piece in the jigsaw puzzle which makes wireless communication possible. You must have used it unwittingly everyday. Chances are, the algorithm embedded in the hardware of your mobile phone already utilises Fourier transform (hence, complex numbers) in some form. Also, your internet browser uses something similar to the Fourier transform called DCT to decode images compressed using JPEG.

I hope this satisfies your curiosity...:)