Arithmetic with Infinite Decimal Expansions
- [Play
Video] 4½ minutes: Algebraic View of Limits. Example
involving sums and quotients.
- [Play
Video] 5½ minutes: Limits and Error Control for Linear
Expressions
- [Play
Video] 2¾ minutes: Error Control to N decimal Places, say 5 or
10.
- [Play
Video] 3¼ minutes: Limits as Error Control for an
unlimited number of decimal places.
Each real number can be regarded as the limit of an
infinite decimal expansion. Arithmetic with real numbers
now requires a discussion of the addition and
multiplication etc of infinite decimal expansions. The
latter will involve some limit concepts and/or the
discussion of the continuity of arithmetic operations
+,-,¸ and ×. The result of these
operations on a pair a and b of real numbers with
infinite decimal expansions can be defined as limit of the
sequence which results from performing the
corresponding operation on the decimal expansions to n
decimal places of each real number a and b, for
n = 1,2,3,¼ and so on.
The technical details are omitted here.
FOOTNOTE: They are
to be
found in the appendices below or in the first chapter in the text
Calculus by Lipman Bers (Holt, Rinehart and Winston 1969,
SBN 03-065240-5). This text
This text was mentioned
earlier.
The details describe say how an error in
the knowledge of two numbers a and b affect the error
in say the (decimal) computation of a+b, a-b, a·b, [1/(b)] and [(a)/(b)] = a ·[1/(b)]. The
omitted details (given in one of the appendices) further imply the algebraic properties of
limits.
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Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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