**Useful books for shoring up mathematical foundations**

A professional hazard in neuroimaging is being years or decades out of formal mathematical training but needing to remember or learn certain mathematical constructs or techniques. Sometimes that works by direct attack, but sometimes it requires working backwards until you are on firm ground conceptually and then working back up. Here are books I personally like for these purposes. You can find them used at Powells, sometimes on Amazon, or Google Books, etc.

__Precalculus Mathematics in a Nutshell__by George Simmons. An as-compact-as-humanly-possible summary of high school math, with generous room in the margins for scribbling. The geometric section is exceptionally well done. Great for seeing what is weak or forgotten from high school.

__Geometry__, and

__Algebra__, by Harold Jacobs. Excellent, well-illustrated, and simple treatments of the basics of the subjects. These textbooks are for children first learning the subject and are widely used in home school curricula. Good for parents wanting to see how their children will learn these subjects, and for anyone wanting to go back to the very beginning of their math education to see what they missed.

__Elements of Algebra__by Leonhard Euler. Why learn algebra from anyone else? An amazing treatment of the subject through cubic and quartic solutions with liberal doses of number theory and Diophantine algebra. The organization is a real surprise, with equal signs not introduced for 60 pages as he first lays out the basis of arithmetic, irrationality, and even complex numbers before ever writing an equation. Good for someone who learned algebra well but who wants to be taken further.

__Elements of Graphing Data__by William Cleveland. For someone who explores data for a living, the challenges of "seeing" the data are a daily concern. That's exactly the topic of this book, both from a perspective of trying to convey the data to an audience in a careful and accurate manner, and from a perspective of the analyst who is trying to understand (and not miss) complex features of a real-world dataset. This book is good for someone wanting an introduction to rigorous consideration of data presentation, and it's a good lead-in to the Bell Labs / Tukey realm of applied mathematics.

__The Princeton Companion to Mathematics__, edited by Timothy Gowers. While all the books above I would recommend to have on paper, this is one that makes sense to have digitally. It is big, and it's not meant to be read linearly or completely. Instead, it's there when you want a lucid, accessible survey of a field, a topic, a mathematician of interest, where in a few pages you can get the "why" of something without all the details. For example, the Fourier Transform is covered beautifully in 4 pages, the Euclidean algorithm in 3 pages, etc. Good for anyone hoping to make sense of tough topics on their own.

**Other resources**

MIT and Cornell and Stanford and many universities have free online courses. These may be good options, depending on the instructor.

The National Museum of Mathematics, on Madison Square Park in NYC, has loads of online programming at all skill levels.

Wikipedia or other internet sources cover many topics, but the clarity and level of presumed familiarity ranges widely. Often not so helpful for learning.

**Some lectures to jog memories about math fundamentals**

001 Greatest common denominators (.pdf)

002 Numbers via infinity (.pdf)

003 Modulo (.pdf)

004 Permutation tests (.pdf)

005 Graphs (.pdf)

006 Ordering (.pdf)

007 Quadratics revisited (.pdf)

008 Quadratics, again (.pdf)

009 Geometry of i (.pdf)

010 Square roots, post-apocalypse (.pdf)