People often ask me this at parties, and I am never able to give a completely satisfactory answer. Part of the reason for this is that it is very difficult to give people an idea of such a broad subject, especially when they think that math is purely about adding and subtracting numbers. People don't realize that things that they take for granted, like being able to listen a CD or DVD, use Google, take digital pictures and transmit them through the Internet, use Yahoo! maps, rely heavily on a broad collection of ideas from the area of applied mathematics.
The question should rather be: what isn't applied math?
Since this would hardly satisfy anybody either, especially some narrow minded people whose exiguous "high school" education has made them believe that there is nothing more to math than numbers, addition, substraction, multiplication and division, or cryptic algebraic expressions without any meaning, I have decided to put together a rough list of applications of mathematics through various fields of science and technology.
In the remainder of this page I will try to convince you that there are some sensible applications of mathematics besides computing your tax returns.
It is hard to decide where to start. Since applied math had traditionally meant solving partial differential equations using finite element methods on a computer, I decided to leave that kind of applications, along with the uncountable number of applications to physics, to the end of the list.
Notice the abundance of links to Wikipedia articles or elsewhere.
To summarize, math has applications to every exact science and every technical field. In fact, it would not be adventurous or pompous to say, that the only way to make a science "exact" is to provide a mathematical foundation for it.
Among the fields that have benefited from math applications, we have, in arbitrary order (see details below):
Since the Greeks, and probably even before, people realized that much of armony could be explained in mathematical terms. To start with, the reason for which two notes belonging to consecutive octaves are perceived as the like notes, i.e. two C's, is that the ratio of their frequencies is a simple as 2:1. To go a bit further, some sets of notes are perceived to sound good together (i.e. as chords) when their frequencies are related to each other as ratios of small integers. For example, in a common Major chord like C, E, Gthe ratio of the frequencies between G and C is 3:2 and the ratio of the frequencies between E and C is 5:4. This always holds for major chords, no matter what the base note is ( in this case it is C).
For more info on this particular application of math to music check out this Wikipedia article on Math of Western music scales
... including numerical computation, cryptography, hash functions, algorithms for searching, sorting, compressing and encoding information (like JPEG, MP3, ZIP), computability theory, simulation, artificial intelligence, automated game playing and decision taking
The list of applications here is almost endless.
I should start by mentioning that computers were first conceived by Mathematicians (John Von Neumann, Alan Turing), long before they could actually be constructed by engineers. These people were responsible for defining and characterizing reasonably well the meaning of the word computable. Some other mathematicians later realized that there were fundamental limits to what computers could do, even if they were "infinitely" fast and had an "infinite" amount of memory.
The question is still open as to whether the same kinds of limits apply to human brains, i.e. is artificial intelligence in the strong sense even possible? Is there a fundamental difference between what a machine can do and what a brain can do? This again brings us into the realm of phylosophy. An interesting book on this subject is "The emperor's new mind" by Roger Penrose.
How many ways are there to tie a necktie?
An easy answer would be to say that there are clearly infinitely many ways since you can always pull it a little bit more or a little bit less and make the knot look different. So we should reformulate the question as :
How many essentially different ways are there to tie a necktie?
Precising what the meaning of essentially different in the last sentence is not trivial to do. Doing that as well as developing a classification of knots is part of what a certain species of Mathematicians do; "algebraic topologists" do.
You will be surprised to find out that there are still at least dozens if not hundreds of essentially different ways to tie a necktie, and Mathematicians have even a developed a systematic symbolic way to describe how to tie the knot.