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.
Why Slopes
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If you like Volume 3 you may also like Three Skills for Algebra , Exponents & Radicals Exactly, complex numbers, Euclidean Geometry , More Calculus and Pattern Based Reason as well. |
Units in Calculations:
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
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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