The Trinity test occurred on a calm morning. Enrico Fermi, one of the observers, began dropping bits of paper about 40 seconds after the explosion; pieces in the air when the blast wave arrived were deflected by about 2.5 meters. From this crude measurement, Fermi estimated the bomb’s yield to be ten kilotons; he was accurate within a factor of two. Although Street-Fighting Mathematics does not address the problem of estimating bomb yields, it gives us a reasonably generic toolbox for generating quantitative estimates from a few facts, a lot of intuition, and impressively little calculus. As one of the reviews on Amazon says, this book makes us smarter.
Street-Fighting Mathematics — the title refers to the fact that in a street fight, it’s better to have a quick and dirty answer than to stand there thinking about the right thing to do — is based on the premise that we can and should use rapid estimation techniques to get rough answers to difficult problems. There are good reasons for preferring estimation over rigorous methods: the answer is arrived at quickly, the full set of input data may not be needed, and messy calculus-based or numerical techniques can often be avoided. Perhaps more important, by avoiding a descent into difficult symbol pushing, a greater understanding of the problem’s essentials can sometimes be gained and a valuable independent check on rigorous — and often more error prone — methods is obtained.
Chapter 1 is about dimensional analysis: the idea that by attaching dimension units (kg, m/s2, etc.) to quantities in calculations about the physical world, we gain some error checking and also some insight into the solution. Dimensional analysis is simple and highly effective and it should be second nature for all of us. Too often it isn’t; my guess is that it gets a poor treatment in secondary and higher education. Perhaps it is relevant that about ten years ago I went looking for books about dimensional analysis and found only one, which had been published in 1964 (Dimensional Analysis and Scale Factors by R.C. Pankhurst). If Mahajan had simply gone over basic dimensional analysis techniques, it would have been a useful refresher. However, he upps the ante and shows how to use it to guess solutions to differential and integral equations: a genuinely surprising technique that I hope to use in the future.
Chapter 2 is about easy cases: the technique of using degenerate cases of difficult problems to rapidly come up with answers that can be used as sanity checks and also as starting points for guessing the more general solution. Like dimensional analysis, this is an absolutely fundamental technique that we should all use. A fun example of easy cases is found not in Street-Fighting Mathematics, but in one of Martin Gardner’s books: compute the remaining volume of a sphere which has had a cylindrical hole 6 inches long drilled through its center. The hard case deals with spheres of different radii. In contrast, if we guess that the problem has a unique solution, we’re free to choose the easy case where the diameter of the cylinder is zero, trivially giving the volume as 36π cubic inches. Many applications of easy cases are simple enough, but again Mahajan takes it further, this time showing us how to use it to solve a difficult fluid flow problem.
Chapter 3 is about lumping: replacing a continuous, possibly infinite function with a few chunks of finite size and simple shape. This is another great technique. The chapter starts off easily enough, but it ends up being the most technically demanding part of the book; I felt seriously out of my depth (it would probably help if I had used a differential or integral equation in anger more recently than 1995).
Chapter 4 is about pictorial proofs: using visual representations to create compelling mathematical explanations where the bare symbols are non-intuitive or confusing. This chapter is perhaps the oddball: pictorial proofs are entertaining and elegant, but they seldom give us the upper hand in a street fight. I love the example where it becomes basically trivial to derive the formula for the area of a circle when the circle is cut into many pie-pieces and its circumference is unraveled along a line.
Chapter 5 is “taking out the big part”: the art of breaking a difficult problem into a first-order analysis and one or more corrective terms. The idea is that analyzing the big part gives us an understanding of the important terms, and also that in many cases we may end up computing few or none of the corrections since the first- or second-order answer may turn out to be good enough. Mahajan introduces the idea of low-entropy equations: an appealing way of explaining why we want and need simplicity in street-fighting mathematics.
Finally, Chapter 6 is about reasoning by analogy: attacking a difficult problem by solving a related, simpler one and then attempting to generalize the result. The example of how many parts an n-dimensional space is divided into by introducing some n-1 dimensional constructs is excellent, and ends up being quite a bit more intricate than I’d have guessed. This chapter isn’t the most difficult one, but it is probably the deepest: analogies between different areas of mathematics can border on being spooky. One gets the feeling that the universe is speaking to us but, like children at a cocktail party, we’re not quite putting all of the pieces together.
Back of the envelope estimation is one of the main elements of a scientist or engineer’s mental toolkit and I’ve long believed that any useful engineer should be able to do it, at least in a basic way. Others seem to agree, and in fact quantitative estimation is a lively sub-genre of the Microsoft / Google / Wall Street interview question. Speaking frankly, as an educator of engineers in the US, our system fails somewhat miserably in teaching students the basics of street-fighting mathematics. The problem (or rather, part of it) is that mathematics education focuses on rigorous proofs and derivations, while engineering education relies heavily on pre-derived cookie-cutter methods that produce little understanding. In contrast, estimation-based methods require strong physical intuition and good judgment in order to discard irrelevant aspects of a problem while preserving its essence. The “rapidly discard irrelevant information” part no doubt explains the prevalence of these questions in job interviews: does anyone want employees who consistently miss the big picture in order to focus on stupid stuff?
In summary this is a great book that should be required reading for scientists and engineers. Note that there’s no excuse for not reading it: the book is under a creative commons license and the entire contents can be downloaded free. Also, the paper version is fairly inexpensive (currently $18 from Amazon). The modes of thinking in Street-Fighting Mathematics are valuable and so are the specific tricks. Be aware that it pulls no punches: to get maximum benefit, one would want to come to the table with roughly the equivalent of a college degree in some technical field, including a good working knowledge of multivariate calculus.