Writers of math books should realize: no matter how many irrelevant pictures you plaster on every page, people are still going to think your math textbook is an arcane tome of black magic. You might as well make it look like a tome of spells, at least that way it’s cool. No, I don’t need to see an expensively-licensed picture of the pod race from Star Wars Episode 1 to motivate me to study derivatives! Fortunately, there is one math textbook which treats the reader like an adult. And it’s awesome. This text is so awesome I almost cried when I discovered it. I’m talking about a deceptively humbly-named work called “Basic Mathematics
Before even cracking the first chapter, I was already surprised by the author’s name. See, most elementary math books are written by people nobody’s ever heard of. But Serge Lang is a household name– at least when “household” means “household of mathematicians”. He’s among the most famous and prolific and talented authors of math textbooks, but the surprising thing is, his books are usually about ridiculous advanced topics you have to go to grad school to even know they exist. That’s why I was shocked to see he wrote about basic mathematics. It’s like if you went to a 6th grade science lab and saw Einstein at the chalkboard.
The most amazing thing about Lang’s venture into high school mathematics: he discusses basic algebra and geometry using the same crisp and precise language that mathematicians use for higher mathematics. It’s a beautiful language and it’s a tragedy that most people will never encounter it, despite years of compulsory education. That would change right away if high schools adopted Lang’s masterpiece.
Just reading through Serge’s exposition of multiplication and addition somehow makes me feel smarter– and I’m a guy who’s been studying math for almost fifteen years now. Most authors seem to find it impossible to talk about solving linear equations without assuming a condescending, downright insulting tone. But not this guy. He makes me think I’m sitting with him talking about math, like we’re peers working together to discover the theorems and laws from scratch.
I began searching for good books on elementary math because I’m always getting email from people asking: “What’s a good book to learn math on my own?” It seems to be a common quandary. There’s so much trash out there, what with universities and public school boards creating an unnatural market where material is churned out to maximize profit with little concern for giving the unfortunate students a good deal. You can’t really even blame the big publishing houses in this: they know that their target audience is only buying the book because they have no choice. But people like me and the readers of my blog, we want to learn this stuff, and learn it well, and we deserve a lot better.
Serge Lang doesn’t BS the reader. Here’s an example passage from the book:
It is a tradition in elementary schools to transform a quotient like
into another one in which the square root sign does not appear in the denominator. As far as we are concerned, doing this is not particularly useful in general. It may be useful in special cases, but neither more nor less than other manipulations with quotients, to be determined ad hoc as the need arises. Actually, in many cases it is useful to have the square root in the denominator. We shall give two examples…
This is a polite slam against the ridiculous “rationalize every denominator” policy pervading high-school math courses. I love you Mr. Lang! The whole book is chock full of goodies like this. For anyone who left the high-school math room a little confused, this is like taking the “red pill” and seeing The Matrix for what it is. And I do mean that literally– Chapter 17 is all about Matrix Algebra 
What Not To Expect From This Book
Definitely avoid this tome if you’re a fan of modern Textbook Art. You’ll find no picture of a motorcycle on the cover. No human pyramid of grinning old people in suits will introduce the next chapter. No gimmicky “Explore and Discuss” exercises– if you have to differentiate discussion-worthy exercises from non-discussion-worthy ones, it’s time to throw away the latter!
Avoid this textbook if you like books written by ten different coauthors. In defense of standard textbook writers, I’m sure every particular one of them has very good intentions. It’s not any single author’s fault when the book goes to committee. When you take ten different visions of perfection and mix them together, you end up with a brownish muck. Mediocre works are churned out by committees, but it takes a lone man to pen a magnum opus.
Don’t read “Basic Mathematics” if you’re hunting for lots of tedious busywork. The exercises focus more on proving things logically rather than computing and crunching numbers. I love the beauty of true mathematical proof, and I’ve always wanted to share it with the whole world. When I read a particularly beautiful proof I want to grab people on the street and say, “Have you heard this!” If only public schools employed a little less Prentice Hall and a little more Serge Lang Hall, a lot more people would get to share this joy of mine!
Buy Serge Lang’s “Basic Mathematics” Now
FURTHER READING
Five Ways to be Better at Math
“Problems” in Mathematics
James Heisig’s “Remembering The Kanji”
How To Train Your Mathematical Maturity