But for people who can read calculus, and sometimes just plain algebra, the drop-dead basic mathematics of a field may not take that long to learn. And it's likely to change your outlook on life more than the math-free popularizations

orthe highly technical math.

Relates the speedup of a sub-task to the resulting speedup of the whole. Trivially true, but often needed to knock down false intuition.

- on Wikipedia, long with examples
- on MathWorld, short without examples

Used to abstract away units and fixed overhead when analyzing resource usage.

- on Wikipedia, long
- cheat sheet from the same article

Traditional square one of theoretical computer science, with many practical applications.

- on Wikipedia, definition and example
- homework with solutions (PDF)

Illustrates many recurring themes. Understanding the proof and usage of the pumping lemma will help you understand and apply more famous, advanced results (e.g. anything involving Turing Machines).

- at Penn Engineering, explanation and examples
- handout (PDF) with concise statement and examples

An astonishingly elegant technique for proving certain kinds of theorems. Originally introduced by the mathematician Georg Cantor to show that the set of real numbers is uncountable – that is, there is no one-to-one correspondence between real numbers and natural numbers, but was later found to generalize to several other contexts. Perhaps the most notable uses of this technique, in addition to Cantor's proof, are Alan Turing's answer to the Halting problem, and Gödel's proof of his famous first incompleteness theorem.

- on Wikipedia, definition and a step-through of the proof
- Halting Problem
- at Michigan State, problem, theorem, and proof
- University of Edinburgh, explanation of proof

**Books**

- How Everything Works: Making Physics out of the Ordinary by Louis Bloomfield (no math though)

**Videos**

- Khan Academy 800+ Youtube videos covering everything from basic arithmetic and algebra to differential equations, physics, and finance

- The Simple Math of Everything by Eliezer Yudkowsky
- Creating The Simple Math of Everything by Matt Simpson, calling for more contributions on the topic.
- Eric Drexler on Learning About Everything by Vladimir Nesov