Full Frontal Calculus:

An Infinitesimal Approach

Single-variable calculus, presented with panache: crisp, concise, and with a smoky infinitesimal flavor.

 

Download the book as a free pdf here.

After reading a bit and finding you like it, be a mensch and purchase a paperback copy at Amazon for $15. 

Other Books

Lobachevski Illuminated

A historical introduction to non-Euclidean geometry developed in the midst of extensive commentary on one the subject's foundational documents: Lobachevski's Theory of Parallels.

 

Winner of the Mathematical Association of America's Beckenbach Book Prize in 2015.

The book can be purchased at the website of the American Mathematical Society

Precalculus Made Difficult

Algebra, Coordinate Geometry, Functions, and Trigonometry.

Learn 'em well! The fate of the free world depends on it!

All of high-school mathematics in one 200-page textbook.

Download it as a free pdf here. If you like what you see, you can support a starving math teacher (or at least one who might have skipped lunch) by purchasing a print copy for $15 at Amazon.

 

Infrequently asked questions

Full Frontal Calculus

Why should I purchase a physical copy when I can download the pdf for free?


The reasons are legion. To take just a few examples:

  1. Holding a physical book in your hands encourages serious, active reading.
  2. A conspicuously displayed copy of Full Frontal Calculus will impress that pretty girl (or handsome lad) in the coffee shop.
  3. 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."
  4. 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.)




Is that an avocado on the cover?


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.




What do you mean by "an infinitesimal approach" to calculus?


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.)





Precalculus Made Difficult

Why should I purchase a physical copy when I can download the pdf for free?


The reasons are legion. To take just a few examples:

  1. Holding a physical book in your hands encourages serious, active reading.
  2. A conspicuously displayed copy of Precalculus Made Difficult will impress that pretty girl (or handsome lad) in the coffee shop.
  3. 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."
  4. 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.)




Why "Made Difficult"?


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.





General

Seth Braver? Who's he?


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.




Is there an email address at which I can contact you?


Indeed there is: bravernewmath@gmail.com