Skip to main content
Engineering LibreTexts


  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Science and engineering, our modern ways of understanding and altering the world, are said to be about accuracy and precision. Yet we best master the complexity of our world by cultivating insight rather than precision.

    We need insight because our minds are but a small part of the world. An insight unifies fragments of knowledge into a compact picture that fits in our minds. But precision can overflow our mental registers, washing away the understanding brought by insight. This book shows you how to build insight and understanding first, so that you do not drown in complexity.

    Less Rigor

    Therefore, our approach will not be rigorous—for rigor easily becomes rigor mortis or paralysis by analysis. Forgoing rigor, we’ll study the natural and human-created worlds—the worlds of science and engineering. So you’ll need some—but not extensive!—knowledge of physics concepts such as force, power, energy, charge, and field. We’ll use as little mathematics as possible—algebra and geometry mostly, trigonometry sometimes, and calculus rarely—so that the mathematics promotes rather than hinders insight, understanding, and flexible problem solving. The goal is to help you master complexity; then no problem can intimidate you.

    Like all important parts of our lives, whether spouses or careers, I came to this approach mostly unplanned. As a graduate student, I gave my first scientific talk on the chemical reactions in the retinal rod. I could make sense of the chemical chaos only by approximating. In that same year, my friend Carlos Brody wondered about the distribution of twin primes—prime pairs separated by 2, such as 3 and 5 or 11 and 13. Nobody knows the distribution for sure. As a lazy physicist, I approximately answered Carlos’s question with a probabilistic model of being prime [32]. Approximations, I saw again, foster understanding.

    As a physics graduate student, I needed to prepare for the graduate qualifying exams. I also became a teaching assistant for the “Order-of-Magnitude Physics” course. In three months, preparing for the qualifying exams and learning the course material to stay a day ahead of the students, I learned more physics than I had in the years of my undergraduate degree. Physics teaching and learning had much room for improvement—and approximation and insight could fill the gap.


    In gratitude to my teachers, I dedicate this book to Carver Mead for irreplaceable guidance and faith; and to Peter Goldreich and Sterl Phinney, who developed the “Order-of-Magnitude Physics” course at Caltech. From them I learned the courage to simplify and gain insight—the courage that I look forward to teaching you.


    For many years, at the University of Cambridge and at MIT, I taught a course on the “Art of Approximation” organized by topics in physics and engineering. This organization limited the material’s generality: Unless you become a specialist in general relativity, you may not study gravitation again. Yet estimating how much gravity deflects starlight (Section 5.3.1) teaches reasoning tools that you can use far beyond that example. Tools are more general and useful than topics.

    Therefore, I redesigned the course around the reasoning tools. This organization, which I have used at MIT and Olin College of Engineering, is reflected in this book—which teaches you one tool per chapter, each selected to help you build insight and master complexity.

    There are the two broad ways to master complexity: organize the complexity or discard it. Organizing complexity, the subject of Part I, is taught through two tools: divide-and-conquer reasoning (Chapter 1) and making abstractions (Chapter 2).

    Discarding complexity (Parts II and III) illustrates that “the art of being wise is the art of knowing what to overlook” (William James [24, p. 369]). In Part II, complexity is discarded without losing information. This part teaches three reasoning tools: symmetry and conservation (Chapter 3), proportional reasoning (Chapter 4), and dimensional analysis (Chapter 5). In Part III, complexity is discarded while losing information. This part teaches our final tools: lumping (Chapter 6), probabilistic reasoning (Chapter 7), easy cases (Chapter 8), and spring models (Chapter 9).

    Finding Meaning

    Using these tools, we will explore the natural and human-made worlds. We will estimate the flight range of birds and planes, the strength of chemical bonds, and the angle that the Sun deflects starlight; understand the physics of pianos, xylophones, and speakers; and explain why skies are blue and sunsets are red. Our tools weave these and many other examples into a tapestry of meaning spanning science and engineering.


    Sharing this Work

    Like my earlier Street-Fighting Mathematics[33], this book is licensed under a Creative Commons Attribution–Noncommercial–Share Alike license. MIT Press and I hope that you will improve and share the work noncommercially, and we would gladly receive corrections and suggestions.

    Interspersed Questions

    The most effective teacher is a skilled tutor[2]. A tutor asks many questions, because questioning, wondering, and discussing promote learning. Questions of two types are interspersed through the book. Questions marked with a ► in the margin, which a tutor would pose during a tutorial, ask you to develop the next steps of an argument. They are answered in the subsequent text, where you can check your thinking. Numbered problems, marked with a shaded background, which a tutor would give you to take home, ask you to practice the tool, to extend an example, to use several tools, and even to resolve an occasional paradox. Merely watching workout videos produces little fitness! So, try many questions of both types.

    Improve our World

    Through your effort, mastery will come—and with a broad benefit. As the physicist Edwin Jaynes said of teaching [25]:

    [T]he goal should be, not to implant in the students’ mind every fact that the teacher knows now; but rather to implant a way of thinking that enables the student, in the future, to learn in one year what the teacher learned in two years. Only in that way can we continue to advance from one generation to the next.

    May the tools in this book help you advance our world beyond the state in which my generation has left it.


    In addition to the dedication, I would like to thank the following people and organizations for their generosity.

    For encouragement, forbearance, and motivation: my family—Juliet Jacobsen, Else Mahajan, and Sabine Mahajan.

    For a sweeping review of the manuscript and improvementsto every page: Tadashi Tokieda and David MacKay. Any remaining mistakes were contributed by me subsequently.

    For advice on the process of writing: Larry Cohen, Hillary Rettig, Mary Carroll Moore, and Kenneth Atchity (author of A Writer’s Time [1]).

    For editorial guidance over many years: Robert Prior.

    For valuable suggestions and discussions: Dap Hartmann, Shehu Abdussalam, Matthew Rush, Jason Manuel, Robin Oswald, David Hogg, John Hopfield, Elisabeth Moyer, R. David Middlebrook, Dennis Freeman, Michael Gottlieb, Edwin Taylor, Mark Warner, and many students throughout the years.

    For the free software used for typesetting: Hans Hagen, Taco Hoekwater, and the ConTEXt user community (ConTEXt and LuaTEX); Donald Knuth (TEX); Taco Hoekwater and John Hobby (MetaPost); John Bowman, Andy Hammerlindl, and Tom Prince (Asymptote); Matt Mackall (Mercurial); Richard Stallman (Emacs); and the Debian GNU/Linux project.

    For the NB document-annotation system: Sacha Zyto and David Karger.

    For being a wonderful place for a graduate student to think, explore, and learn: the California Institute of Technology.

    For supporting my work in science and mathematics education: the Whitaker Foundation in Biomedical Engineering; the Hertz Foundation; the Gatsby Charitable Foundation; the Master and Fellows of Corpus Christi College, Cambridge; Olin College of Engineering and its Intellectual Vitality program; and the Office of Digital Learning and the Department of Electrical Engineering and Computer Science at MIT.

    • Was this article helpful?