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Tutors - All Subjects use or become a tutor at your own risk YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
Chapter 20
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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.
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. |
The chapter Arrow Addition describes addition of arrows with or on maps, with parallelograms, with perpendicular components and with coordinates. This chapter provides some background, optional reading, for the next two chapters on complex numbers.
The chapter Complex Numbers - Basic Ideas begins
the description of the geometric approach. It explains the
polar-coordinate add the angles, multiply the lengths
rule for calculating the product of two points or
arrows in the plane.
FOOTNOTE: If you liked, you could explain the ideas in all but the last section of this chapter to someone who only knows (a) how to measure the polar and rectangular coordinates of points; (b) how to use both kind of coordinates to locate points in the plane; and (c) how to add real numbers and also how to multiply positive numbers.This description of complex numbers confirms or derives the law of sign for real numbers, and shows that the square root of -1 is easily thought of.
The chapter Complex Numbers - Links to Trig describes how the add the angles, multiply the length polar-coordinate based multiplication implies the standard expressions of the product in terms of the rectangular coordinates, alias real and imaginary parts, of the factors. This chapter show how the addition and multiplication of arrows is linked to the angle sum identities for the sine and cosine functions. The link justifies the simplified, algebraic approach to trigonometry favored by science and engineering students.
The last chapter Complex Logs, Powers and Exponentials
gives formulas for logarithms,
exponentials and powers of complex numbers. These formulas
are expressed simply in terms of the logarithms, exponentials,
powers, sines and cosines of the rectangular and polar
coordinates of complex numbers. The latter are regarded
as points in a coordinate plane.
FOOTNOTE: Other texts more properly label the rectangular coordinates of a complex number, its real and imaginary part. Other texts will call the length and angle polar coordinates of a complex number its magnitude (or modulus) and argument.This provides a connection with trigonometry and the earlier discussion of complex numbers. The formulas are of service in engineering, science, statistics, etc.
The above provides a high school level story, to explains the basic properties of complex number. The foregoing ends this story except for the last chapter below. It describes logarithm, exponentials and related functions of complex numbers z = x+iy. The link to trigonometry reappears there.
www.whyslopes.com
Volume 3, Why Slopes and More Math -Foreword, One Calculus preview and Online Chapters: (V) signals video (RealPlayer Format) to watch
Area Entrance & Hub Foreword Chapter Descriptions 1. Introduction 2. Calculus Starter Lesson 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 Content Guide
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.
Online Volume 2, Three Skills for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills and concepts, those needed in calculus, again to make the hard easier. A visual understanding of complex numbers may help - serve as back ground info, in partial fraction decomposition.
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