- Taschenbuch: 152 Seiten
- Verlag: The Mit Press; Auflage: New (13. April 2010)
- Sprache: Englisch
- ISBN-10: 026251429X
- ISBN-13: 978-0262514293
- Größe und/oder Gewicht: 17,8 x 0,6 x 22,9 cm
- Durchschnittliche Kundenbewertung: 2 Kundenrezensionen
- Amazon Bestseller-Rang: Nr. 95.373 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving (Englisch) Taschenbuch – 13. April 2010
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"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
"'Too much mathematical rigor teaches rigor mortis.' This is Sanjoy Mahajan's way of saying 'failure to make timely approximations leads to algebraic paralysis.' Approximations are essential in the design process, and his book legitimizes, accelerates, and extends the methods that students eventually have to learn on their own anyway."--R. David Middlebrook, Professor Emeritus of Electrical Engineering, California Institute of Technology
""Street-Fighting Mathematics" taught me things I wish I'd learned years ago. It's fun, fast, and smart. Master it and you'll be dangerous."--Steven Strogatz, Cornell University, author of "The Calculus of Friendship"
"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
Über den Autor und weitere Mitwirkende
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.
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So startet das Buch denn auch mit dem Konzept der Dimensionsanalyse, welches kreativ angewandt wird um Integrale zu lösen. Die Methoden werden generell jeweils immer an illustrativen Beispielen eingeführt und erprobt. Später wenn die Probleme schwieriger werden verfolgt der Autor das Motto "When the going gets tough, the tough lower their standards." Folglich opfert man etwas Genauigkeit, kann dafür aber die Probleme überhaupt erst lösen (oder erspart sich zumindest umständliche Rechnungen welche nur zu einem geringfügig anderem Resultat geführt hätten).
Es gibt Bücher, die zwar von sehr intelligenten Menschen geschrieben worden sind, aber trotzdem kaum lesbar und verständlich sind. Hier jedoch hatte ich den Eindruck dass die Faszination und Genialität des Autors zum Leser hindurchscheint und ansteckend ist. Mittlerweile erwische ich mich nämlich des Öfteren dabei, anstatt den Rechner zu benutzen Resultate lieber von Hand näherungsweise auszurechnen, was viel lehrreicher ist (und sicherer, zumal man dann weiss, woher das Resultat genau kommt anstatt blind dem Taschenrechner zu vertrauen).Lesen Sie weiter... ›
Just take the examples as pars-pro-toto: You may not be really interested in easily guessing formulae for areas inside ellipses. But the technique of reduction to easy cases always helps me to be sure what I am talking about. So this book is worth five stars.
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The book is reminiscent of Consider a Spherical Cow by Harte, The Art and Craft of Problem-Solving by Zeitz, and How to Solve It by Polya. Although much of the book focuses on how to avoid doing integrals and taking derivatives, it presumes the reader is familiar with calculus. In this respect it's different from the books I just mentioned and other ones out there on approximation, e.g. Guesstimation. The example problems are diverse, most are borrowed from physics, geometry, and math, with a few are that are Fermi-type "real-world" scenarios.
My main complaint is that the book is so short. I wish the author had combined this book with material from his Order of Magnitude Physics and Art of Approximation courses, which are marvelous not only for the problem-solving on display but also for the physics content.
This type of book addresses a serious gap in American math-science education. Learning techniques for approximation allows one to tackle the sort of ill-posed problems one is most likely to encounter in the real-world. It is also intimately tied to recognizing the salient features of a problem, such as the physical principles involved in a physics problem or the most questionable assumption in an economic model. Street-Fighting Math deserves a wide readership and will hopefully influence other math-science teachers and authors.
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 36' 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 as PDF. 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.
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!
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.