Karl's Calculus Tutor

Infinitesimals, Hyperreals, Nonstandard Analysis

Nonstandard Analysis and the Hyperreals, by Jordi Gutierrez Hermoso

Here, hyperreals are an extension (superset) of reals, and are defined by sequences of real numbers.  (An infinitesimal number is a hyperreal whose absolute value is less than that of any real number.  You can think of it as a hyperreal whose "closest" real number is zero.)  For example, 0.999... can be construed to represent the hyperreal number given by the sequence

(0.9, 0.99, 0.999, ...)

and you can take 1 to mean the real number given by the sequence

(1, 1, 1, ...)

Then 0.999... is not equal to 1. Their difference 1-0.9r is the infinitesimal number

(.1, .01, .001, ...)

which is larger than zero but smaller than every positive real number.

Elementary Calculus: An Approach Using Infinitesimals

On-line Edition, by H. Jerome Keisler

This is a calculus textbook at the college Freshman level based on Abraham Robinson's infinitesimals, which date from 1960. Robinson's modern infinitesimal approach puts the intuitive ideas of the founders of the calculus on a mathematically sound footing, and is easier for beginners to understand than the more common approach via limits.

The First Edition of this book was published in 1976, and a revised Second Edition was published in 1986, both by Prindle, Weber & Schmidt. The book is now out of print and the copyright has been returned to me as the author. I have decided (as of September 2002) to make the book available for free in electronic form at this site. These PDF files were made from the printed Second Edition.

The whole book in one large file (24 megabytes)

Single chapters in much smaller files:

Preface to First and Second Editions

Contents and Introduction

Chapter 1 Real and Hyperreal Numbers

Chapter 2 Differentiation

Chapter 3 Continuous Functions

Chapter 4 Integration

Chapter 5 Limits, Analytic Geometry, and Approximations

Chapter 6 Applications of the Integral

Chapter 7 Trigonometric Functions

Chapter 8 Exponential and Logarithmic Functions

Chapter 9 Infinite Series

Chapter 10 Vectors

Chapter 11 Partial Differentiation

Chapter 12 Multiple Integrals

Chapter 13 Vector Calculus

Chapter 14 Differential Equations

Appendix