Mathematics – behind the scenes of everything

 kw: book reviews, nonfiction, mathematics, applications

After a forty-year career as a scientific programmer, AKA "coder", I can look back to see that I was primarily a working mathematician. The scientists whose methods I embodied in computer code were, of course, having the computer "do the math", but I frequently had to correct their math. They were all brilliant, but one cannot always expect someone whose life has been devoted to chemical engineering or mineralogy or seismic analysis to have kept up their math skills over the prior couple of decades. On the other hand, I greatly enjoyed calculus and other "mid level" math operations, so I was "up" on what they needed and could make sure they used the math properly. I don't claim to understand perhaps 90% of the higher level math in the current literature. But I understand enough that I could make a career of it. 

In all that time, I developed only a few new methods, and published only a single peer-reviewed article, to be found at Science Direct. The abstract is open. Sadly, the article is behind Elsevier's paywall. But the key takeaway is this: I had to develop new methods to numerically solve the very stiff differential equations used by physical chemists studying the conversion of organic grunge (they call it kerogen) into crude oil. Relevant to the current book, I used methods called "convergence acceleration", which were developed before crude oil was a thing. In particular, one method was first used to study stresses in earthen dams, and another was used by Leonard Euler in the mid-1700's, for a project I don't now recall. I borrowed a couple of related methods from a theoretical dissertation by a colleague at my graduate school.

What's the Use?: How Mathematics Shapes Everyday Life, by Ian Stewart, a retired Professor of Mathematics who has at least five times my expertise, is based on a notion first expressed by Eugene Wigner in a 1960 article titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

Wigner was not remarking on math's broad effectiveness. That isn't hard to understand. Rather, mathematicians and others who use lots of math find that methods, perhaps derived for specific problems, or perhaps for their theoretical beauty, are found to be useful in realms so remote that it seems miraculous. As the author points out, some say, "The Universe must be made of mathematics!"

The book starts off with a brief historical survey, reaching back far beyond Leonard Euler. However, Euler is responsible for a breakthrough in complex analysis that led to a formula, called Euler's Identity, which displays the essential unity of all mathematics:

The five symbols, here related by two operators (the "+" and the "="), are combined into an astonishing expression. Let's unpack them, from right to left:

• 0, zero: Before the year 1200AD, the zero as a placeholder had been in use for about 500 years, but was not yet accepted as a number, outside of India and China. Only in the 1700's (in Europe) were zero and the negative numbers accepted as numbers, making subtraction, for example, immensely more useful.
• 1, one: The first of the "natural numbers" or "counting numbers" is the original number.
• π, pi (pronounced "pee" in Greek, but most of us say "pie"): This is the ancient symbol for the ratio of the circumference of a circle to its diameter. Millennia of effort to "square the circle" were based on the belief that π is a rational number (one that can be expressed as the ratio of two natural numbers; 335/113 is a useful approximation, but is not exact). Only in the 1700's was it proven that π is an irrational number, which is expressed by a string of digits that never ends and never repeats. Being related to the circle means it is the basis of trigonometry, but that is only the beginning!
• i, the "imaginary" number: This is the square root of minus one. It has no place in any of the hierarchy of "number line" numbers: natural numbers, integers, rational numbers, and irrational numbers, which together constitute the "real" numbers. The combination of a real number and some real-number multiple of i is a complex number. Complex numbers became useful when it was realized that they represent coordinates in the plane.
• e, Euler's number: This was originally the base of natural logarithms, which show up in the solutions to many calculus problems. It is named for Euler, but was actually assigned by John Napier a century earlier, when he developed natural logarithms. Its value is approximately 2.7182818285… e and π are the first two irrational numbers to be proven to be transcendental, which has an esoteric meaning related to polynomial derivations. Many (infinitely many) irrational numbers are the solutions to polynomial equations, but most (more infinitely many!) are not. However, they are hard to find. Natural logarithms and their inverse, exponential expressions, are found everywhere in both calculus and complex analysis.

The hard part, which seems magical to many, is to evaluate e^ix, where x is some real number, and then to show that when x = π, the expression's value is -1. Endnote 50 in What's the Use? is a very short proof that exponentiation with i becomes a rotation, meaning a trigonometric combination: e^ix = Cos(x) - i*Sin(x). When x = π, the Sin part equals 0 and the Cos part = -1. This is the connection to π.

Why is this important? Much trigonometric algebra is much easier to carry out in this form. The operations automatically keep track of all the Sin and Cos functions that are embedded in the exponential expressions. Electrical engineering, frequency analysis, and a host of other disciplines would be either impossible or a great deal more difficult without complex analysis using exponential expressions.

What does this have to do with everyday life? Cell phone communications use digital decomposition and reconstruction of audio signals. Getting the digital signals transmitted efficiently requires some high-powered math. Turning a song into an MP3 file, so it takes up 1/10th or 1/20th the space on your hard drive (or phone memory) is a several-step mathematical exercise. Doing the same with a visual image to produce a JPG file is similar, and the five steps, drawn from five quite diverse realms of mathematics, are described—in brief!—in Chapter 10, "Smile, Please!".

Before getting to that point, however, the author discusses efforts to allot voting districts "fairly", describing several definitions of "fair", along with at least some hints of a proof that no matter what you may call "fair", it can't be done perfectly. He discusses the relationship between a problem involving seven bridges and two islands, that is actually insoluble, but is related to equitable ways to allocate kidneys for transplants, which is soluble. The way encryption works in your web browser (and email, I hope!) and your phone is based on "trap door functions" which are, of course, mathematical in nature. He also shows ways being developed to make much stronger trap doors to cope with the immense computing power that quantum computing just might deliver. Then, we have Einstein's theories of relativity (there are two, Special and General): both are needed to get GPS to function accurately, in addition to several other realms of mathematical operations.

There are 13 chapters showing that math is hidden behind a great deal of what goes on in the world. Civilization is impossible without it. In case this fills you with dread, remember that you don't have to be an automotive engineer to drive a car, but we do need some automotive engineers to have cars to drive. Thus, not all of us have to understand higher math to use our GPS, cell phone, or microwave cooker, but there need to be some pretty bright mathematicians out there to make these things work.

Don't shy away from this book because it is about mathematics. The author's writing is very readable, and he does his best to help us glimpse the way some of these things work. One book won't make much of a dent in your struggles with algebra, or calculus, or whatever. But it will yield an appreciation for the unreasonably diverse ways almost any mathematical development could be used for practical things later on.