- Series: MIT Press
- Paperback: 152 pages
- Publisher: The MIT Press; New edition (March 5, 2010)
- Language: English
- ISBN-10: 026251429X
- ISBN-13: 978-0262514293
- Product Dimensions: 7 x 0.2 x 9 inches
- Shipping Weight: 8.8 ounces (View shipping rates and policies)
- Average Customer Review: 23 customer reviews
- Amazon Best Sellers Rank: #167,022 in Books (See Top 100 in Books)
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Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving (MIT Press) New Edition
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In problem solving, as in street fighting, rules are for fools: do whatever works--don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation.In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge--from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool--the general principle--from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest. Street-Fighting Mathematics grew out of a short course taught by the author at MIT for students ranging from first-year undergraduates to graduate students ready for careers in physics, mathematics, management, electrical engineering, computer science, and biology. They benefited from an approach that avoided rigor and taught them how to use mathematics to solve real problems.Street-Fighting Mathematics will appear in print and online under a Creative Commons Noncommercial Share Alike license.
"Many everyday problems require quick, approximate answers. Street-Fighting Mathematics teaches a crucial skill that the traditional science curriculum fails to develop: how to obtain order of magnitude estimates for a broad variety of problems. This book will be invaluable to anyone wishing to become a better informed professional."--Eric Mazur, Balkanski Professor of Physics and of Applied Physics, Harvard University
"All students and teachers of mathematics and science, whatever their level, will find a wealth of fun and practical tools in this fantastic book." David MacKay, Fellow of the Royal Society, Professor of Natural Philosophy, Cavendish Laboratory, University of Cambridge, Chief Scientific Advisor, UK Department of Energy and Climate Change
About the Author
Sanjoy Mahajan studied mathematics at the University of Oxford and received a PhD in theoretical physics at the California Institute of Technology. He is now Associate Director of the Teaching and Learning Laboratory and a Lecturer in the Department of Electrical Engineering and Computer Science at MIT. Before coming to MIT, he was a Fellow of Corpus Christi College, Cambridge, and a Lecturer in Physics in the University of Cambridge.
Top customer reviews
Especially the chapter 1, "Dimensions", I liked, because always I on the dimensions for a helping control against failures, and of cause it's always for the finishing result. And this also as result of during my engineering education then having a professor that talked too low and besides on the wall table made awful drawings of the Greek letters, so that I always had to try with connecting guessed dimensions until I found out which Greek letters he actual had tried to write. And all of the teachers only seldom used time for putting on dimension at all!
But absolutely an excellent mathematical book which I started reading as soon as I received it and continued during the following days when having possible time. And absolutely a book in which I in the future again will bee reading, for example just for relaxing.
Many engineers have a lot of training in formal mathematics, but for the most part we have much less training in creative problem solving of mathematics. The author starts off with a classic problem from elementary physics and shows how it can be solved (or at least approximated) by dimensional analysis, and from there proceeds to detail numerous problem solving techniques including the use of easy cases, pictures, and analogy. Each chapter is accompanied by several detailed examples explaining the technique in question, as well as numerous exercises (usually difficult ones) requiring the use of a given technique. The author uses examples from many fields, but especially ones from geometry, calculus, classical mechanics, and fluid mechanics. Some highlights from each chapter.
Chapter 1: Dimensional Analysis
Estimating the form of the Gaussian Integral. Finding the form of the free-fall equation from ballistics.
Chapter 2: Easy Cases
Analysis of the drag force (using dimensionless groups). Volume of a frustrum.
Chapter 3: Lumping
Estimating integrals. The spring-mass differential equation. Navier-Stokes and the Reynold's Number
Chapter 4: Pictorial Proofs
Shortest bisector of a triangle. Summing series.
Chapter 5: Taking out the big Part
Mental Multiplication/Division. "Entropy" of expressions
Chapter 6: Analogy
Bond angle of methane. Euler-MacLaurin summation.
As you can see, the examples are varied, and tend to be things that are commonly seen (and struggled with) in many fields. The author shows how the general technique can be used many different ways, while also allowing the reader to figure out some problems on his or her own.
This book is an essential read for any scientist or engineer who should definitely have the tools presented in their repertoire. I am giving this book 4 stars only because sometimes the mathematics in the example problems gets a little bit confusing, and some of the ordering of the sections in the chapters is a bit muddled. In addition, there's really little reason to buy a print version of the book, considering there is a free version (albeit not exactly the same) on the MIT open courseware website along with additional problems (with solutions).
In short, I thoroughly recommend this book to scientists, engineers, and anyone with a somewhat more than casual interest in mathematics; however you are probably better off reading the free online copy rather than buying the print version.
To not make things sound further oblique, Richard Feynman learnt the wonders of using the Liebniz's integral rule from a book by Woods, Advanced calculus when he was relegated to the back of the class to peruse it, as he was too 'bored' with the mundane (I'm assuming) plug and chug of high school calculus. He describes it in his witty account Surely You're Joking, Mr. Feynman! as a bag of tools he would use repeatedly to solve integrals other graduate students were stumped by.
If you think I'm taking this anecdotal analogy too far, you need look no further than the first chapter of street fighting math. Starting with a warm-up on dimensional analysis for free-fall, where Mahajan gives us a hint of heuristics methods in store, he jumps into some guesswork for evaluating the Gaussian integral.
If you have gaped at the wizardry of a calculus teacher performing that pirouette to polar coordinates and effectively increasing the complexity of the problem to crack it, wait till you see what Mahajan does. Without giving away how the postman did it, he culls ideas from diverse areas as dimensional analysis and synthesizing simple results to bear fruit on the solution.
The second chapter delves into the mystery of our thinking process, or - if you might prefer to call it - our fumbling process. When in doubt, try it out! Go big, go small - see if it meets the expected picture. Also, what is often missed in textbooks or even in class, is what mistakes can teach us, sans the cliche. To be intentionally led to barking up the wrong tree might sound like a waste of time to the hurried, but when you're trying different things and recognise it as a natural course of events, you will be left wondering how honest it is to actually have it spelt out in a book like this. I had never anticipated that the volume of a pyramid would open up such rich generalizations, proofs and destruction of intrinsic geometric myths. The crowning glory, is, however, the fluid mechanics problem of drag. I would strongly suggest a video at a TED conference which is available on YouTube where Mahajan performs the simplest and most graceful demonstration. It also takes dimensional analysis to the next level, so to speak, because it introduces the idea of dimensionless groups, which has its roots in the work of a certain Dr. Buckingham, who wrote a paper about it in 1914.
A small digression here. Mahajan seems to have the uncanny ability to separate the wheat from the chaff, and brings to bear his diverse historical knowledge of the development of physics education. He does this not for any sense of chronology or authenticity, but from a direct relevance to present problems. Growing up in India under the aegis of the British System, stumbling across things like fundamental and derived quantitites and dimensional arguments was akin to discovering a sunken galleon with a rich past. Personal reminesences aside, the idea of reducing dependendent variables to dimensionless sets has considerable relevance to analysing physical situations. Reducing complexity may often make a daunting problem tractable and weed out spurious dependencies, as Pankhurst points out in his book, Dimensional Analysis and Scale Factors (page 82)
Chapter 3 deals with those long forgotten graphs to demonstrate what integration and differentiation really are. Summing and tangentiating, so to speak. When in trouble (with an area under a nasty curve), box your way out, seems to me the motto, a fitting manifesto for the street-fighter. Areas under curves are tackled with two kinds of heuristics and applied to Stirling's formula. The approximation is so close and its resemblance with the Gaussian so uncanny, how can we not guess the connection to the exact result?
Differentiation is moved from the realm of limits and evanescent algebraic quantities and formulaic drivel to the geometry of sketching secants, better secants, and when that doesn't work, finding what does, which is estimating functions. There is some inspired guesswork in applying these ideas to the harmonic oscillator. The real sleight of hand, however, is the introduction of the 'correction factor' when extrapolating to a finite amplitude case. We get a result, pick over its vulnerabilites, compensate for what we can, and come up with a better guess. Lo and behold!
Chapter 4 is for what I call the imaginative mathematical artist, because it requires you to stop pushing symbols, and learn estimation by sketching. Following Mahajan's cue, I would strongly recommend the books by Nelson, Proofs without Words: Exercises in Visual Thinking,Proofs Without Words II: More Exercises in Visual Thinking as well as Math Made Visual: Creating Images for Understanding Mathematics. Again, this is an appeal to our dormant geometric sense. There is an amazing Archemedian development of finding a curve that minimizes length by successive approximation.
Chapter 5 is where I learnt to do arithmetic again, after all these years. It is also where I began to underline the book because things were getting sublime. That there is a connection between information theory and plausible alternatives seems almost a comment on the brevity and economy of expressions. As a case in point, an example shows just how ugly a quadratic solution can actually be, and how much more insight successive approximations can reveal!
The last chapter possibly requires that leap of imagination into the abyss, as it deals with analogies to extrapolate the hitherto unknown, and often, unseeable. How many volumes do 6 planes divide space? Even Martin Gardner would balk. Dr Mahajan guides us gently, though, through thick and thin. I have never been quite as comfortable associating algebraic properties to operators as I have after reading the ebb and flow between discrete and infinitesimal sums. I am still recovering from the way tangent roots have been mollified by taming a bunch of polynomials. While I admit to cheating with the Basel sum by looking up approaches on the web (you'll stumble across the solution immediately, so I suggest you don't do it), I take consolation in the fact that the jump earned Euler his first laurels. However, Mahajan gets complete credit for making historically rich problems seems within the bounds of mere mortals. Never have I felt quite as much remorse as I have after being so close to the solution. My advice: stick with every problem in the book - its absolutely worth the effort.
Some problem highlights:
2.27 on low reynold's number (the reference to Purcell's paper at the back of the book is an eye-opener. It goes to show how methods in this book could be applied to far-flung areas like locomotion of organisms in fluids)
4.26 is on mentally manipulating figures. The answer itself can provide some direction.
4.31(c) requires some astute algebraic judgement.
5.17 on quadratic approximation requires exploiting symmetry.
5.28 on interpreting a diagram is the art of isolating and expanding the important terms
6.30 calculating the exact Basel sum would be an aha feeling, or more!
In short, as the author puts it, we are here not to analyse things to death, but only to the extent that they are useful or tractable. This might seem anathema to purists, but the humbling fact remains that some approximation techniques are often more elegant, and sometimes, surprisingly, the only way out! You need to come to this book with the least of preconceptions, for then, your unlearning will be most effective.
I cannot say this for many books - but I will admit that this reads like a thriller. I worked through it uninterrupted except for pacifying skirmishes that broke out between my 4 and 5 year old. I took it to bars, much to the disbelief of my friends. I even destroyed an ereader which stored the book, riding a roller-coaster with it at Legoland. For that alone, I have earned a Darwin award.
May you use the many damn tools in this wonderful book again and again!
Most recent customer reviews
Wish Sarah SIlverman had written it.
She has more STREET SMARTS.