**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.

Find your weak spots:

__Pre-calculus in a Nutshell__by George Simmons.

Easy reintroductions:

__Geometry__by Harold Jacobs.

__Algebra__by Harold Jacobs.

__Trigonometry: A Clever Study Guide__by James Tanton.

Meaty extensions of classical geometry and algebra:

__Elements of Algebra__by Leonhard Euler.

__Elements__by Euclid.

__A sequel to the first six books of the Elements__

__of Euclid__by John Casey.

Calculus:

__Calculus with Analytic Geometry__by George Simmons.

__Calculus__by Tom Apostol.

Set theory, infinity, construction of number systems:

__Fundamentals of Mathematics__by Bernd Schroeder.

__Naive Set Theory__by Paul Halmos.

__Set Theory__by Felix Hausdorff.

Modern Algebra:

__A Concrete Approach to Abstract Algebra__by W.W. Sawyer.

__Algebra__by Michael Artin.

__Introduction to Linear Algebra__by Gilbert Strang.

Reference:

__The Princeton Companion to Mathematics__, edited by Timothy Gowers

**Comments**:

If you haven't read many math books, you may think that one book on a subject is as good as another. That is definitely not true.

There are many styles of math book, and you will want different styles at different stages of understanding. Here are some styles. One is the pop math book, for example those by Steve Strogatz (

__Infinite Powers__), Jesse Ellenberg (

__Shape__), and others. The main purpose of these books is to pique curiosity in the lay public, not to teach. I don't mention these kinds above. A second type is a math text written for a lay audience, but whose purpose is to teach. Authors like W.W. Sawyer (

__Mathematician's Delight__) and N.Ya. Vilenkin (

__Stories about Sets__) work more in this mode. These kinds of books have depth, often have a handful of exercises, and often have solutions or work through an exercise just after posing questions. These are good for the self-studier, as they often provide plenty of motivation and commentary on a topic or technique. Usually these books are focused (W.W. Sawyer's

__A Concrete Introduction to Abstract Algebra__), but they can even be full-bore texts hundreds of pages long. As an example: in his old age, Leonhard Euler went blind, and required an assistant for daily life. On learning that his assistant did not know algebra, to remedy the situation, Euler dictated

__Elements of Algebra__. The book starts from nothing, builds number systems, and over hundreds of pages passes far beyond high school algebra, all in a conversational style intended for the interested, intelligent layperson. A third kind of math book is the workhorse textbook, something we have all encountered in mathematical training. They teach, but the student may not take much joy in the learning. Contrast such textbooks with a fourth kind of textbook that addresses the same topics but in ways that are especially interesting in the exposition, historical grounding, the kinds of problems posed, etc. Nearly all the books listed here belong to this category, in part because I work off of recommendations of others, and in part because I don't include what I didn't like unless it filled a gap I otherwise couldn't fill. Especially notable here is the work of George Simmons (

__Calculus__) and Paul Halmos (

__Naive Set Theory__). Simmons brings topics alive with his seemingly encyclopedic knowledge of the development of mathematics, and his personality shines through vividly in each of his books. Halmos writes in a nearly deceptively simple and conversational manner about complicated topics, conveying them in his own, unique manner. Importantly though, Simmons clearly has the "bottom-up" view and writes to teach the newcomer, while Halmos appears to be writing to the newcomer, but is actually reframing hard, advanced material in unadorned language.

In general, I've only put books here I have thoroughly used and enjoyed. Also, I only put books that serve core purposes. If a person is happy with their mastery of this material, they certainly don't need my suggestions for their next steps.

It is no good to read a text and not work its exercises. The exercises in a good book are not merely about developing "muscle memory". Authors of good books choose and sequence exercises carefully in order to prompt realizations and insight.

Finally, reading a math text takes time. A paragraph may take a minute, an hour, or a week to grasp. That's normal. If you find yourself stuck, look at other books on the topic. They will have other takes, and one of them will give you the key you need.

**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.

**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)

011 Null hypotheses (.pdf)