www.whyslopes.com > Volume 3, Appendices,,  Decimal View of Real Analysis,  1995  >   Content Guide    

Decimal View of Real Analysis

The appendices in Volume 3 deliver decimal-based proofs of the main theorems in calculus and real analysis. This decimal tack may be sufficient for people not specializing in mathematics while providing students of pure mathematics, a context or motivation for the decimal free proofs stemming from set-based, decimal-free codification of mathematics.

Have a good look at the above table. Then note most of this site area has been duplicated and posted online in the new Calculus Introduction site area.

The Appendices -- Beyond Calculus
 - Real Analysis, a Decimal Intro and Context

Appendix A. What's Next points to two references that could be read besides this work.

Appendix B (poster material) gives finite and infinite version of the Pigeon Hole Principle, and then employs the infinite version  to give a decimal-based proof of the Bolzano-Weierstrass Theorem. Here we assume that a real number is defined by an infinite decimal expansion or an infinite (recursive) process that in principle, if not practice, determine more and more digits in that expansion without stopping.

Appendix C (poster material) covers the Triangle Inequality as is and Generalized

Appendix D defines and shows the existence of greater lower bound and least upper bounds for bounded Sets & Sequences. Then it considers the limits of bounded  Monotonic Sequences.

Appendix E gives or states Error Control Inequalities needed or used to derive the Algebraic Properties of limit.

Appendix F begins with a theorem to identity What Functions are Continuous. The theorem is to be used recursively. For continuous functions, appendix F continues with statements and proofs of theorems

• Closed Range Theorem
• Intermediate Value Theorem
• A Finite Covering Theorem - Compactness in other words.
• Equicontinuity Theorem
• Extreme Value Theorems

Appendix G covers the properties of differentiable functions:

• Rolle's Theorem - A technical result needed for the next theorem.
• Mean Value Theorem. - the key theorem for differentiable functions.
• Constant Difference. Theroem - why two functions, differentiable everywhere on an interval, with the same first derivative on that interval, differ by a constant.
• Lipschitz Continuity - How a bounded derivative implies Lipshitz continuity.
• Velocity Revisited - how bounded velocity implies Lipschitz continuity for functions which describe how position depends on time.
• Sufficient Conditions

Appendix H cover integration theorems

• Riemann Sums Convergence - a proof or two of the fundamental theorem of calculus in general and for base-2 Riemann Sums, based on the Equicontinuity Theorem for  continuous integrands.
  
• Lipschitz Continuous Integrands and  convergence of Riemann Sums. See  how a bounded derivative implies Lipshitz continuity.

Some instructor favor the latter approach for first courses in calculus.

Postscript:  One Sided Range Theorems (online only)

Related Chapters in Volume 3

Chapter 14 Limits & Error (poster material) takes a technical turn. It starts with the decimal viewpoint of Limits, Error Control and Continuity. Sections cover the limits of functions, jumps in functions, Cauchy Sequences, Significant Digits, and limits of sequences from the decimal error control viewpoint of if there a guarantee that a computation will give a limit to finite or unlimited number of decimal places. Motivation is thus provided for the standard decimal-free ed viewpoint of limits and continuity.

The appendices, cover still more advanced material. They  give proofs normally omitted in calculus and provide a decimal view of limits, continuity, convergence and compactness. The Foreword describes the origins and motivations for this work and some of its chapters.

Chapter 15 What is Slope (poster material)provides motivation for the standard approximation for the slope (or derivative) of a nonlinear function and then says if the approximation converge to a limit, that limit is the slope (or derivative). That is a  twist of the typical kind in calculus or its interpretations.  Saying how to compute or approximate a quantity, here the slope (or derivative) in the limit, defines it if there is convergence.

Chapter 16 What is Velocity (poster material) provides motivation for the standard approximation for the velocity   non-constant speed motion,  and then says if the approximation converges to a limit, that limit is the velocity (a slope or derivative).  That is a second twist of the typical kind in calculus or its interpretations.  Saying how to compute or approximate a quantity, here the velocity in the limit, defines it if there is convergence.

Chapter 17 What is Area (poster material) provides motivation for the standard approximation for the area of a region  and then says if the approximation converges to a limit, that limit is the area.  That is a third twist of the typical kind in calculus or its interpretations.  Saying how to compute or approximate a quantity, here the area in the limit, defines it if there is convergence. Area here is approximated by covering with squares.

The first section of Chapter 18 Integration (poster material) moves from approximating areas under a curve y = f(x) > 0, a special region,  with squares to a covering with rectangles and thus introduces the Riemann Sum approximation. The second second shows or suggest how to compute the limit, assuming it exists through  anti-differentiation,  that is a reversal of the slope computation process. The anti-differentiation process is justified in some cases by the Second fundamental theorem of calculus. The assumption that the limit of area approximations exists is justified by the first fundamental theorem of calculus. The proof of the latter is given in the last appendix as an appetizer for advanced students.

The middle section of Chapter 18 introduces a function F(x) by defining it as the area under a curve y = f(x) between two points a and x. The last section of chapter 18 describes five more ways to define functions, ways often met in mathematics before, in and after calculus.

Chapter 19 Logs & Powers introduces or defines the natural logarithm, another function,  as the (signed) area under a curve. This definition is similar to that which appears in the middle section of Chapter 18, the discussion or derivation of the second fundamental theorem of calculus. Further logarithms are defined (briefly, too briefly) in terms of the natural logarithm, while the exponential function and one non-negative real number raised to a power (another real number) are introduced ro defined with the aid of the inverse to the natural logarithm.

Vectors and Complex Numbers Revisited.

Chapter 20 What's Next describes the next chapters.

Chapter 21 Add Vectors shows how to add vectors in the plane, and attempts to provide motivation for it. The presentation here is an exploration of ideas which I wanted to clarify for myself, if not the reading.

Chapter 22 Complex Numbers (Basic Ideas) introduces this numbers geometrically. 

Elements of this chapter appear in the simplified treatment of complex numbers (& trig), posted online at this site after the completion of Volume 3.

Chapter 23 Complex Numbers (Links to Trig) first points to the well-known simplification of trigonometry given by the use of complex number function cis(A) = cos(A) + i sin(A) = exp(iA) in higher mathematics and the mathematical disciplines, engineering and physics included. 

The last part of Chapter 23 Complex Numbers (Links to Trig) explores different ways to establish the complex number,  distributive law for multiplication over addition. The simplified treatment of complex numbers (& trig), posted online at this site after the completion of this book, Volume 3 is simpler. But I am still puzzled regarded the optimal way to develop complex numbers and trigonometry impurely from a mix of assumptions about arithmetic and geometry. In pure  mathematics, trigonometric functions may be defined without reference or dependence on geometric diagrams.  However, novices need diagrams of one kind or another for their first comprehension of trig functions.

Chapter 24 Complex Logs  etc states formulas for logarithms, exponentials and powers of complex numbers, and formulas for the hyperbolic functions.  It provides no more information about the functions.

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