6.055J | Spring 2008 | Undergraduate

The Art of Approximation in Science and Engineering


Singing Logarithms and Physical Constants

Calculating logarithms using musical intervals (PDF)

Constants for back-of-the-envelope calculations (PDF)

Required Text

Updated Version (2014)

The notes used in this course have now been enhanced and published in book form by The MIT Press. The complete book is available as a free download from the OCW website:

The book is also available in paperback for purchase:

  • Buy at MIT Press Mahajan, Sanjoy. The Art of Insight in Science and Engineering: Mastering Complexity. MIT Press, 2014. ISBN: 9780262526548.

As Used in This Course (2008)

The course readings were based on an early draft of the above published book. Those course readings are provided below as a single file, or broken down in the subsequent table by discrete topics for each class session. Chapter 3 in Part 1 is unavailable.

Complete course notes (PDF - 1.2 MB)

Readings by Session

Readings given in the table below were designed to be read after that session’s discussion and are taken from the manuscript above.



1 Preview (PDF)

Managing complexity (PDF)

Pit spacing (PDF)

Managing complexity:

Heterogeneous hierarchies:
2 Tree representations (PDF)

Oil imports (PDF)

3 Gold or bills? (PDF)
4 Random walks (PDF)

Random walks for accuracy (PDF)


Removing spurious complexity:

Proportional reasoning:
6 Triangle bisection (PDF)
7 Pentagonal heat flow (PDF)

Introduction plus spring mass period (PDF)

Mountain heights (PDF)

8 Animal jump heights (PDF)

Calculus (maximizing a quadratic) (PDF)

Jump heights (PDF)

9 Calculus (maximizing gain of a second-order system) (PDF)

Power limits to jumping (PDF)

10 Falling cones (PDF)

Calculus (maximizing a quadratic) (PDF)

11 Drag and jumping (PDF)
12-14 Cycling, swimming, flying (PDF)
Conservation/box models:
15 Conservation cube (PDF)
16 Flight (PDF)
17 Flight (scaling) (PDF)
18 Flight (explicit range and speed) (PDF)
Dimensions and afterword:
19 Algebraic symmetry (PDF)
20 Economics example; pyramid example (PDF)
21 Dimensionless groups (PDF)

Hydrogen size (PDF)

22 Densities (PDF)

Hydrogen size by physics (PDF)

23 Long-lasting learning 1 (PDF)
24-25 Bending of light (PDF)
26-29 Long-lasting learning 2 (PDF)

Buckingham Pi Theorem (PDF)

Discarding actual complexity:

Special cases:
30-31 Pyramid volume (PDF)

Pulley acceleration (PDF)

32-34 Drag! (PDF)
35-36 Random walks and keeping warm (PDF)
Spring models:
37-38 Waves (PDF)

Course Info

Learning Resource Types
Problem Sets with Solutions
Online Textbook