Braver New Math

An infinitesimal is, by definition, an infinitely small number - smaller than every positive real number, yet still greater than zero. No real number satisfies this definition, so infinitesimals are not real. And yet, by developing a "calculus of infinitesimals" (as the subject was known for most of its first two centuries), mathematicians and physicists achieved unparalleled insight into real functions, breaking through the static algebraic ice shelf to reach a flowing world of motion below, changing and evolving in time. Calculus is, among other things, the mathematics of continuous change. But even as calculus ushered in modern science, philosophical problems remained. Do infinitesimals really exist? Or are they just useful fictions? And how could mathematics - of all subjects - afford the risk of fictional foundations? Such problems were not resolved until the 19th century, when mathematicians developed a rigorous way to do calculus without infinitesimals. This new basis for calculus (the theory of limits) put the philosophical questions to bed, but it also made calculus less intuitive for the average user, who was not, after all, a philosopher, and who rather liked the feel of infinitesimals. Full Frontal Calculus returns infinitesimals to the stage, restoring the intuitions that motivated the creation of the subject, and putting you, the reader, in touch with the thoughts of the intellectual giants who developed it. You'll end up learning all the same results as someone who studies a more typical limit-based approach to calculus, but your path there will be more intriguing, simpler, and considerably shorter. (Most textbooks devote 500+ pages to developing single-variable calculus. In FFC, this requires just 160 pages.)

The reasons are legion. To take just a few examples: Holding a physical book in your hands encourages serious, active reading. A conspicuously displayed copy of Full Frontal Calculus will impress that pretty girl (or handsome lad) in the coffee shop. Physical books absorb (and later, return) some of your experiences in ways that e-books do not. Years later, when you take Full Frontal Calculus down from your bookshelf, its old familiar feel in your hands will stir dormant memories. As you read again the old notes in the margins scribbled by your younger self, and see again the odd coffee stain or spatter of tomato sauce, you'll find yourself murmuring, "Yes. I remember, I remember." When you buy a physical copy, I will have the satisfaction of seeing its sale on Amazon. You, in turn, will have the satisfaction of having caused my satisfaction. With the royalties I earn from your copy, I'll be able to afford a cup of chai. When I drink it, I'll think of you. (And you'll have the pleasure of thinking of me thinking of you as I drink my chai.)

No. It is a small stone, which is precisely what the word "calculus" means in Latin. (May God protect you from renal calculi.) Our distant ancestors used pebbles (calculi) as aids for simple calculations. Descendants of those humble pebble manipulations included the abacus, positional notation, and all that follows... including the subject we now call "calculus". Isaac Newton famously described himself thus: "I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me." Among the smoothest of Newton's calculi was... calculus.

Patience, patience. FFC may yet sprout chapters on multivariable calculus, but for now you must settle for some pandemic-era videos on the topic that I made while teaching online. Consider them a sneak preview.

I'm afraid my attempts to enlist Morgan Freeman's assistance to make an audiobook version of FFC have proved fruitless, and I will settle for nothing less. Aut Freeman, aut nihil. That said, there are supplementary videos covering Chapters 1 through Pi (differential calculus) and Chapters 7 and 8 (series, parametric equations, polar coordinates) that I made for my students. Should you seek succor in the sound of my voice or the sight of my handwriting, these videos should satisfy your peculiar, albeit highly refined, desires.

My approach is similar in spirit to 3Blue1Brown's famous "Essentials of Linear Algebra" video series on YouTube, but rather than offering a video overview of the subject's essentials, The Dark Art of Linear Algebra is a complete textbook. As such, it includes basic topics not covered in the 3Blue1Brown series (especially the computational side of linear algebra), and of course, it contains lots of exercises. The Dark Art emphasizes visual, visceral understanding, rooted in the familiar geometry of n-dimensional Euclidean space. (Familiar, that is, when n = 2 or 3, but by the book's end, you will feel much more at home in spaces of 4 or more dimensions.) This emphasis of visual intuition leads me to introduce linear maps before matrices and matrices before Gaussian elimination, reversing the order in which these topics appear in most textbooks. Determinants, change-of-basis, eigenstuff, projections, and least squares solutions to inconsistent systems are all presented geometrically, too.

I understand your disappointment. You wish to learn linear algebra, a noble and difficult pursuit. I am saddened by the thought of talented young people having to forgo their linear algebraic studies for want of $25 to buy a paperback copy of my book. Society, I firmly believe, should smooth the path for budding scholars such as yourself, easing their financial burdens whenever possible. But I am not Society. I am one man trying heroically to support multiple mistresses while making payments on two condos, not to mention the Braver New Math yacht. And cocaine is not getting any cheaper. Every dollar counts, and if I can find a way to extract a few more from you, then by God I intend to do it.

The text itself is about 150 pages, yet it covers all the standard topics in an introductory linear algebra course. Of course, the title page, table of contents, index, answers, copyright page, and so forth add a bit more bulk to the finished product. When all is said and done, DALA's handsome covers surround precisely 185 pages. I have it on the authority of Goldilocks herself - a keen student of mathematics - that the length of The Dark Art of Linear Algebra is just right. A student can read the entire book, cover to cover, in a single college class.

Because mathematics is difficult, and it was made that way. Not by me, but by nature. Mathematics is also supremely logical. One can master it by understanding why it works. Understanding is hard work - it is difficult. It is also rewarding. Accordingly, I've written this book with two Albert Einstein quotations in mind: one for my readers, and one for me. "Any fool can know. The point is to understand." "Make everything as simple as possible, but not simpler." Too many precalculus books - especially the "made easy" variety - try to make the subject "simpler than possible", thus conveying (at best) mere knowledge without understanding. Precalculus Made Difficult does no such thing. It emphasizes understanding throughout, and assumes, with a spirit of generosity rare among contemporary textbooks, that you, the reader, have the will to learn well and the ability to read closely and carefully.

The reasons are legion. To take just a few examples: Holding a physical book in your hands encourages serious, active reading. A conspicuously displayed copy of Precalculus Made Difficult will impress that pretty girl (or handsome lad) in the coffee shop. Physical books absorb (and later, return) some of your experiences in ways that e-books do not. Years later, when you take Precalculus Made Difficult down from your bookshelf, its old familiar feel in your hands will stir dormant memories. As you read again the old notes in the margins scribbled by your younger self, and see again the odd coffee stain or spatter of tomato sauce, you'll find yourself murmuring, "Yes. I remember, I remember." When you buy a physical copy, I will have the satisfaction of seeing its sale on Amazon. You, in turn, will have the satisfaction of having caused my satisfaction. With the royalties I earn from your copy, I'll be able to afford a cup of chai. When I drink it, I'll think of you. (And you'll have the pleasure of thinking of me thinking of you as I drink my chai.)

The main difference is that it is now published by Vector Vectorum Books, that most exclusive - and secretive - of publishers. To celebrate Precalculus Made Difficult's new home, it now sports a stylish new cover. The content is substantially the same as the 1st edition, but some of the original's infelicities of typesetting have been resolved, numerous typos were corrected, and its exposition and exercise sets have been discreetly polished here and there.

He's an old hand at teaching mathematics, which he has done in 10% of the United States. Among the many places he has lived, his favorite is probably Missoula, Montana, where he earned his Ph.D. (UM, 2007). Since 2010, he has taught at South Puget Sound Community College in Olympia, Washington.

Indeed there is: bravernewmath@gmail.com

I do, I do. While I tune my guitar before playing my next piece, please feel free to toss your fivers, ten spots, C-notes, and higher right here.