In teaching a mainstream Calculus I course, many if not most of us are content to soft-pedal any serious discussion of the mathematical justification for such things as the limit definition of the derivative. I, for one, promote my real analysis course in part as “the rest of the story” (with apologies to Paul Harvey, of course), with the explanation that we’ll get a chance to examine why manipulations that lead us to 0/0 or a casual use of infinity wind up working out.
There is, of course nothing wrong with deferring serious questions of rigor to a later course. It’s consistent with the historical development of calculus, and it allows us to focus on the important uses of calculus in that first course, which is entirely appropriate. Anthony Gardiner has conceived Understanding Infinity as “an introduction to why the calculus works” much as we promise to deliver in real analysis, but it’s something more than that. There’s a very clear discussion, under the provocative heading “What’s Wrong with the Calculus?” that lays out the issues involved — including why it took the mathematical community so long to confront these issues — but in and among the expected explanations of derivatives and integrals, there are explanations of several other mathematical ideas involving infinity.
A big reason why this book achieves its declared purpose so well is that it’s not a real analysis textbook. Freed of the requirements of that format, the author is free to travel through a variety of interesting mathematical topics that look at infinity from a casual perspective and then proceed to place those ideas on a rigorous foundation. Infinite decimals are used to motivate infinite series, and we then return to calculus with a look at Fourier series. Area and volume are approached from first principles, without a derivative or integral in sight, and before long, the author is describing the challenge of two-dimensional sets without area. The books ends up with a full explanation of the concept of “function”, and so we return to the issues raised in elementary calculus.
There’s a lot of interesting mathematics here that’s out of the mainstream of the “calculus to real analysis” transition, and that ultimately makes this book well worth reading.
Mark Bollman (email@example.com) is associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.
|PART I FROM CALCULUS TO ANALYSIS|
|CHAPTER I.1 What's Wrong with the Calculus?|
|CHAPTER I.2 Growth and Change in Mathematics|
|PART II NUMBER|
|CHAPTER II.1 Mathematics: Rational or Irrational?|
|CHAPTER II.2 Constructive and Non-constructive Methods in Mathematics|
|"CHAPTER II.3 Common Measures, Highest Common Factors and the Game of Euclid"|
|CHAPTER II.4 Sides and Diagonals of Regular Polygons|
|CHAPTER II.5 Numbers and Arithmetic?A Quick Review|
|CHAPTER II.6 Infinite Decimals (Part 1)|
|CHAPTER II.7 Infinite Decimals (Part 2)|
|CHAPTER II.8 Recurring Nines|
|CHAPTER II.9 Fractions and Recurring Decimals|
|CHAPTER II.10 The Fundamental Property of Real Numbers|
|CHAPTER II.11 The Arithmetic of Infinite Decimals|
|CHAPTER II.12 Reflections on Recurring Themes|
|CHAPTER II.13 Continued Fractions|
|PART III GEOMETRY|
|CHAPTER III.1 Numbers and Geometry|
|CHAPTER III.2 The Role of Geometrical Intuition|
|CHAPTER III.3 Comparing Areas|
|CHAPTER III.4 Comparing Volumes|
|CHAPTER III.5 Curves and Surfaces|
|PART IV FUNCTIONS|
|CHAPTER IV.1 What Is a Number?|
|CHAPTER IV.2 What Is a Function?|
|CHAPTER IV.3 What Is an Exponential Function?|