Books on quick calculation methods
On mobile devices, read the page in horizontal/"landscape" mode. Otherwise math may not fit the screen.This is a list of books that contain quick and dirty calculation methods: ways to get answers to physical or mathematical questions quickly and unrigorously. Fermi problems, like the famous piano tuner problem, are good examples. I’m thinking of a more broad category, though – a mixture of numerical estimation, guesswork, symmetry arguments, dimensional analysis, and so on. Here’s a list of such books and a quick review of each of them.
If you want the most bang for your buck (or time!), I recommend Guesstimation 1 and 2 for numerical estimates and Fly by night physics for solving physics problems quickly.
Here’s the list:
- Guesstimation 1 and 2
- Fly by night physics
- Qualitative analysis of physical problems
- Street-fighting mathematics
- The mathematical mechanic
- Secrets of mental math
- Modern physics from an elementary point of view
Guesstimation 1 and 2
These are basically Fermi problem books. The books first go through some general methods and then give dozens of examples in each book of how to apply the methods.
For example, the books teach you to take geometric averages instead of arithmetic averages (tend to be closer to the real answer if you know an upper and lower limit). You can take a geometric average approximately by taking two values, for example \(5\cdot 10^7\) and $7\cdot 10^9$, and averaging the power of ten and the number in front of it. For example, average of 5 and 7 is 6. For the powers of ten the average of 7 and 9 is 8. So the geometric average of those two numbers is \(6\cdot 10^8\). An example of a problem considered in the books is how many feet of toilet paper does a household use in a year.
That, and other tricks, can be learned from the book. The best part is that it mostly consists of examples, which contain hints to the solution. If you want to learn how to give off quick numerical estimates, these are the best books for practicing.
Suitable for all levels. No prior mathematical background is needed.
Fly by night physics
Fly by night physics: how physicists use the backs of envelopes by A. Zee.
This book contains quick and dirty methods for solving physical problems and is my favorite physics book of all time. It forces you to look at physics homework-style problems and solve them without caring for rigor, without bothering to apply a fancy theory. As the subtitle says: solving physics on the back of an envelope.
The book tells you how to estimate the yield of a nuke from pictures, makes you match wits with Einstein over diffusion, and shows you how Bohr derived his famous atomic model. The problems are solved by using e.g. physical intuition, dimensional analysis and symmetry.
I thought that the early parts of the book were somewhat better than the later. In the last chapter, Zee deals with particle physics, but he was perhaps undone by his own expertise; I don’t think you’d get much out of it unless you were already familiar with the material.
It’s nevertheless a great book. I would recommend it to any first year physics student or even an enthusiastic highschool student. It’s the ultimate source of dubious physical arguments. Suitable for undergraduates and above.
If you’d like to see me derive the Maxwell-Boltzmann speed distribution this way, see here.
Qualitative analysis of physical problems
Qualitative analysis of physical problems by M. Gitterman.
This book covers much of the same methods as Zee’s book, but in a systematic, “academic” fashion. It shows you, from the beginning, how to construct models of physical situations, and then solve them using dimensional analysis, perturbation theory and symmetry arguments.
If you’re looking to cover similar methods as “Fly by night physics”, but want a more systematic approach of slightly more mathematical sophistication, this is the book.
Suitable for undergraduates, though some material may be more advanced.
Street fighting mathematics
Street fighting mathematics is to math what Fly by night physics is to physics. It contains methods for estimating integrals without actually calculating them, pictorial proofs, reasoning by analogy, and so on.
The first chapter, for instance, teaches you how to calculate integrals by inventing dimensions for the variables involved.
It’s a fun book, and “opportunistic problem solving” captures its essence well.
Suitable for undergraduates in physics/mathematics.
Also, I toyed around with the worst way to calculate definite integrals, just in case you doubt that I am a real street-fighting math user.
The mathematical mechanic
The Mathematical Mechanic: Using Physical Reasoning to Solve Problems.
This is a fascinating book that can reasonably be read out of order. It’s essentially a collection of solutions of mathematical problems by physical reasoning.
It’s best to illustrate by example: the book calculates the integral
\[I=\int _0^1 \frac{x}{\sqrt{1-x^2}}dx\]by considering a mass of weight $P=1$ mounted on a frictionless vertical track on a string of length 1.
If that sounds like something you’d be interested in, this is the book. The methods don’t very easily generalize; the book is best enjoyed as a collection of clever tricks to admire, rather than a textbook on how to solve problems.
Best suited for people proficient with undergraduate mathematics.
Secrets of mental math
Contains tricks for doing mental math – things like multiplication, division, sums and so on – in your head.
For example, here’s one trick: how to square a two digit number ending in 5. For example, 35. Take the first number and add one, \(3+1=4\). Multiply these together, \(3\cdot 4 = 12\). Write 25 after it: we have \(35^2=1225\), which is correct.
That trick is specialized, but the book teaches you a bunch of them and how to invent your own. If you want to learn how to calculate quickly (and impress all your friends at parties!), this is the book.
The book is suitable for anyone, no background in mathematics needed.
Modern physics from an elementary point of view
Modern physics from an elementary point of view by V. Weisskopf.
These are lecture notes by V. Weisskopf and go over atomic theory, nuclear theory and some astrophysics theory using, as the title implies, elementary methods.
For example, in the first chapter Weisskopf derives binding energies for atoms using very simple arguments.
This text is for people who are already familiar with the fancy theory, but want to learn to look at it from a simpler point of view. If you don’t know anything about atomic theory, it may simply be confusing.