Author | Jerome H. Keisler | Entered | 2004-01-08 08:58:02 by Ryanov |

Edit | edit data record | Freedom | Copylefted, but with restrictions on modification and/or sale (disclaimer) |

Subject | Q.A - Mathematics. Computer science (analysis) | ||

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I wish I'd learned calculus from it!by Ben Crowell (crowell09 at stopspam.lightandmatter.com (change 09 to current year)) on 2005-01-26 14:41:04, review #441 http://www.lightandmatter.com |

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Textbooks are usually unoriginal, because most teachers are conservative in their choices. They get used to teaching a subject a certain way, and don't want to change. This is a calculus textbook with a very unusual approach. It was published in 1976, and evidently was successful enough, despite its idiosyncracy, to justify a second edition a decade later. Its publisher, however, eventually allowed it to go out of print. The copyright has reverted to the author, and he has made it available in digital form on his web site. The digital book consists of pages scanned in from a printed copy and assembled into an Acrobat file, so it's a big download, and you can't do some things with it, such as searching the text for a particular word.

The title leaves no doubt that the book is different. Whereas most textbooks these days define derivatives and integrals in terms of limits, this one uses infinitesimals. The real numbers are generalized to make a numer system called the hyperreal numbers, which include infinitesimally small numbers as well as infinitely large ones. Essentially, this represents a return to the way Newton and Leibniz originally conceptualized the calculus, but with more rigor.

I don't know about other people, but when I learned calculus, I got very
uneasy when we got to the Leibniz notation. My teacher said that dy/dx
wasn't really one number divided by another, but rather an abbreviation
for the limit of the quantity Δy/Δx. That wasn't so bad, but
what really made me queasy was when he then suggested that you could usually
get the right answer by treating these dx and dy thingies *as if* they
were numbers. The scary part was that word "usually." What was legal and what
wasn't? How many sizes of infinitesimals were there? Was it legal to
say that 1/dx was infinite?
What operations would lead to paradoxes? What about proofs that used
infinite numbers to show that 1=2?
The wonderful thing about this book is that you end up knowing exactly
what you can and can't do with infinities and infinitesimals, and you get
to use the Leibniz notation in all its intuitively appealing glory.
For instance, the chain rule really can be proved simply by writing
(dz/dy)(dy/dx)=dz/dx, simply canceling the dy's.

It would be interesting to see how students reacted to this book when learning calculus from scratch. I suspect that they'd have an easier time with many of the concepts like implicit differentiation, which seems so awkward in the traditional approach, but they might be scared a little by the initial development of the hyperreal number system. The book develops the hypperreal system axiomatically, which left me yearning for more of a constructive method. Then again, we develop the rational and real numbers axiomatically in high school, so maybe it's not such a big issue. My initial unease was cleared up by a few crucial examples:

- If H and K are infinite, then H-K may be infinite or finite -- it depends on
*which*infinite numbers H and K are. - If H is infinite, then (2H+1)/(H+1) isn't equal to 2, but it differs infinitesimally from 2.
- (H+1)
^{1/2}-(H-1)^{1/2}is infinitesimal.

I confess, however, to a little residual indigestion at the way the author develops the integral. He introduces finite Reimann sums first, and gives several numerical examples. But next, instead of taking the limit of sums with more and more terms, he takes the finite sum with n terms, and replaces n with an infinite integer. Instant vertigo!

This is a wonderful, original textbook.

*Information wants to be free, so make some free information.*

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A very interesting bookby dradetsky on 2004-11-07 17:25:10, review #431 |

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writingbetter than 80% |

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The contents of this web page, except the parts contributed by members of The Assayer, are copyright (c) 2000 by Benjamin Crowell, and are copyleft licensed under the Open Publication License 1.0, without options A or B. |