Interesting real life applications of elementary mathematics

Question

If you teach mathematics to future highschool teachers, you often feel that they are bored because what they learn at university does not have much to do with what they will have to teach in school, and what they will have to teach in school will be boring for their students because it has nothing to do with real life. To remedy this situation, we sometimes offer a course "Mathematics in Real Life" to explain how often mathematics "just happens" in real life, but goes unnoticed. Some sample topics in that course are

• GPS as an application of 3d Euclidean geometry,

• politics: Arrow's paradox and apportionment algorithms

• Existence and meaning of Nash Equilibria as an application of the Brouwer fixpoint theorem

• Compact disks: the sampling theorem and error correcting codes

• MP3 and the modified discrete cosine transform

• pocket calculators and the CORDIC and BKM algorithms

• number theory and cryptography: RSA and discrete logarithms

• Benford's law and accounting fraud detection

I am looking for more examples like those above. They should be accessible to undergraduates (some of the topics above are actually a bit too hard), have some impact on real life, and (ideally) be somewhat surprising for someone who hasn't heard of them yet. Thus, I am neither looking for recreational mathematics as in this question nor for "typical" applications of maths in computer simulation, statistics or finance, which usually involve some deeper mathematics.

Answer 1

Bayes theorem is pretty useful in health care. If 5% of people have cancer and we have a test that is 95% accurate at detecting cancer, what's the chance of you having cancer if the test comes back positive? It turns out that even though the test is 95% accurate, we're still only 50% sure that you have cancer. That's because if we take the number of "True Positives" over the total number of positives, it's only 50% of them.

Answer 2

Symmetry groups have applications to chemistry (crystal structure) and molecular chemistry, to technology and engineering, to architecture, and even to Reactor Calculations, although this is less popular nowadays.