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YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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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
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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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.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
Chapter 22
Complex Numbers - Basic Ideas
Here is a geometric story which describes the complex numbers, or what mathematicians since Gauss in the 1840's have regarded as the complex numbers. This geometric story provides a confirmation of the law of signs for real numbers. This first part of the story could be explain to someone familiar with arithmetic and the measurement of coordinates, rectangular and polar in the plane. The second part of this story is an exercise in trig. The second part appears in the next chapter.
This applet (online only) illustrates the ideas in this chapter.
This first half assumes you have some familiarity with the measurement of distances and angles, with the addition of real numbers and points in the plane, and finally with multiplication of nonnegative (that is zero and positive) real numbers.
1. The immediate motivation for this approach (in this chapter) stems from three successive 1976 McGill University public lectures of the late Richard Feynman. He simply described physics as the addition and multiplication of arrows in the plane. He defined their multiplication as follows: add their angles and multiply their lengths. In terms of the polar coordinate [r1,q1] and [r2,q2] for the factors, the polar coordinates of the product is [r,q] = [r1r2,q1+q2] .)All this was effectively presented to a general audience with no mention of vectors nor the Gauss-Argand representation of complex numbers. See the foreword.
2. In Morris Kline's three-volume work Mathematical Thought from Ancient to Modern Times, see in volume 2, Chapter 27, the third section called The Geometrical Representation of Complex Numbers. This section briefly describes the approach of Caspar Wessel (1745-1818). Part of Wessel's work (translated into English) is reproduced in David Eugene Smith's 1929 work A Source Book in Mathematics, Dover 1959 Reprint.
Points in the Plane
AdditionThe sum of two points with the rectangular coordinates (a,b) and (c,d) is given by (a+c,b+d). We therefore write
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In words, the addition rule is simple add the rectangular coordinates of the summands to get the rectangular coordinates of the sum. With this in mind, the following question is easy: What are the rectangular coordinates of the sum of (1,14) and (2,8)? Answer: (1,14)+(2,8) = (1+2,14+8) = (3,22). The chapter Arrow Addition discusses the addition of points or arrows in the plane further.
Multiplication
Next we define using polar coordinates the product of two points in the plane. Each point or factor is located by means of angular displacement or rotation from the positive real axis, and also a nonnegative distance from the origin. The product of two points is given by a third point. Its angular displacement is the sum of the angular displacement of the factors. Its distance to the origin is the product of the distances of the factors. This is the add the angles and multiply the lengths rule. In polar coordinate notation, the multiplication rule and definition is indicated by
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Example. Two arrows are to be multiplied. One has length 1.3 and angle 22.62°; the other factor has length 1.026 and angle 46.97°; and so their product has length 1.3338 = 1.3·1.026 and angle 69.69° = 22.62°+46.97°; and that is it. See the following diagram.
Another Example. The product of the two points [3,80°] and [4, 60°] is [(3)(4), 80°+ 60°] = [12,140°]
Stop For a Summary
The addition of points in the plane is given by means of their rectangular coordinates while multiplication is given in terms of polar coordinates. It is an exercise in trig to obtain expressions for the rectangular coordinates of a product in terms of the rectangular coordinates of the factors. See the next chapter. (Note that while your reading of this chapter requires a mastery of the algebraic way of writing and thinking, you could explain with diagrams and examples only, the ideas in this chapter to someone without this mastery. It is the on-paper communication which requires the mastery.)
Introducing the Complex Numbers
Points in the plane with the operations of addition and multiplication just given are called the complex numbers. The plane with these two operations on its points is called the complex numbers plane, or more briefly the complex numbers.
We will now change to a more standard notation for them. We may and often will write the rectangular coordinates z = (a,b) as z = a+ib, We will further call the abscissa a, the real part of the complex number z = a+ib. We will also call the ordinate b, the imaginary part of the complex number z = a+ib.
We will say that the complex number z = a+ib is purely imaginary when its real part a = 0. The angle of a purely imaginary complex number z = a+ib = 0+ib = (0,b) is 90 degrees or 270 degrees (modulo
FOOTNOTE: Two quantities x and y are equal modulo a third quantity c, if and only if their difference x-y = kc for some whole number or integer k.360 degrees), depending on the sign of the imaginary part b. When b > 0, the angle is 90 degrees (modulo 360 degrees). When b < 0, the angle is 270 degrees (modulo 360 degrees).
We will also say that z = a+ib is (purely) real when its imaginary part b is zero. The angle of a (purely) real complex number z = a+ib = a+i0 = (a,0) is 0 degrees or 180 degrees (modulo 360 degrees), depending on the sign of the real part a. If a > 0, this angle is 0 degrees (modulo 360 degrees) while if a > 0, this angle is 180 degrees (modulo 360 degrees).
Real Numbers as Complex Numbers
Each complex number z = a+i0 with imaginary part zero gives and is given by a real number a. We will write z = a in this situation, and say that the complex number z is also a real number.
With this practice, the real numbers can be regarded as a subset of the complex numbers; and the real number line can be identified with the horizontal axis of the plane.
Confirmation of The Law of Signs
We identify the real number line with the horizontal axis of the plane. With this identification, observe that positive numbers have angular displacement zero, modulo 360 degrees. Also observe that negative numbers have angular displacement 180 degrees, modulo 360 degrees. The magnitude of a real number is its distance to the origin.
Suppose z = a+i0 and w = c+i0. We
want to compute the product zw with the multiply the lengths, add the
angles rule. Each factor has length |a|
or |c|. Each factor
has angle 0 or 180 degrees (modulo 360 degrees). The relationships
- (+1) = (+1)(+1) as 0° = 0°+0°
- (-1) = (+1)(-1) = (-1)(+1) as 180° = 0°+180° = 180°+0°
- (-1)(-1) = (+1) as 360° = 180°+180°
For the second example, the number -2 is identified with the point (-2,0) = [2,180°]. See the figure below.
Now multiplying the point [2,180°] by itself leads to the product [2,180°]2 = [22,180°+180°] = [4,360°] = [4,0°]. Thus the point on the horizontal axis identified with -2 when squared gives the point identified with +4 indicated above. The 360 degrees in the diagram for the number or point 4 = (4,0) represents the doubling of the angle 180 degrees.
For an example or exercise, compute the pairwise products of 3=3+0i, 4=4+0i, -3=-3+i0 and -4=-4+0i using the add the angles, multiply the lengths rule.
Remark. The add the angles, multiply the length rule could be
used to define the product of real numbers to people/students who know (a) about
the addition of real numbers or coordinates and (b) about the multiplication of
non-negative numbers. They would not need to have any previous knowledge of the
law of signs.
FOOTNOTE: The add the angles, multiple the lengths rule for the multiplication of complex numbers thus yields a rule for the multiplication of real numbers once the multiplication of positive numbers with themselves or zero is understood/defined.
More Exercises. Compute the following using the multiply the lengths, add the angles rule:
- A = (1.5)·(2).
- B = (1.5)·(-2).
- C = (-1.5)·(-2).
- D = (1.5)·(-2).
- E = [10,45°] ·[[1/20],15°].
Stop For A Summary. The polar coordinate definition
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Square Root of -1
The real number -1 = -1+0i = [1,180°] has angle 180 degrees (mod 360 degrees) and length 1. The purely imaginary number (0,1) = 0+i1 = [1,90°] has angle 90 degrees and length 1. Multiplying this point or number by itself, that is, squaring it, gives the point with length 1 ×1 = 1 and angle 90°+90° = 180°. So the product equals -1+0i = -1. We call i = the principal square root of -1.
A second square root of -1 is obtained as follows. The imaginary number (0,-1) = 0+i(-1) = [1,-90°] has angle -90 degrees and length 1. Multiplying this point or number by itself, that is squaring it, gives the point with length 1 times 1 =1 and angle (-90°)+(-90°) = -180° = 180° (mod 360°). So this product equals -1+0i = -1 as well.
This provides two square roots of -1 as both [1,+90°]2 = [1,+180°] = -1 and [1,-90°]2 = [1,-180°] = -1.
Square Roots of Other Complex Numbers
The square root of a positive number or zero are real nonnegative numbers. I assume in the following that you know how to compute these square roots. The square roots of negative numbers and of other arrows or points in the coordinate plane depend on this ability.
Observe that squaring points in the plane doubles their angular displacements and squares their magnitudes (distance to the origin). That is, the add the angles, multiple the lengths rule gives
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Exercises.
- Find all the square roots of 4 and -4 and plot them.
- Find the cube roots of 27 and -27 and plot them in the plane.
- Find the square roots of \cis(45°) = cos(45°)+isin(45°) = [1,45°].
Complex Conjugates
The complex conjugate of a complex number z = a+b i with polar coordinates r = Ö[(a2+b2)] and q is the complex number [`(z)] = a-b i with polar coordinates r and -q.
* Exercise. Show multiplying a complex number a+b i by its conjugate a-b i gives the nonnegative number r2 = a2+b2.
Conjugates and Reciprocals
Observe that p = [(a)/(r2)]-i[(b)/(r2)] = [1/(r2)][`(z)] has angle -q and length [1/(r)]. Here p = [1/(r2)][r,-q] = [[1/(r)],-q].) Multiplying number p = [[1/(r)],-q] by z = [r,q] gives the complex number [1,0] with length 1 and angle 0, that is, the real number 1. And multiplication of any point (c,d) by 1 = [1,0°] yields back the point (c,d)
The reciprocal (or multiplicative inverse) of the complex number z = a+b
i with length r > 0 and angle q is
the complex number p with length 1/r and angle -q.
Observe that if r > 1 then the length of the reciprocal [1/(r)] < 1 < r, that is, the length of the reciprocal is less than 1 and the length of the original number. In contrast, if 0 < r < 1 then [1/(r)] > 1 > r. Question: Which of these two cases is represented in the above diagram? What happens in the case r = 1?
Two Algebraic Properties
Observe
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Real Multiples of Arrows
We said earlier (in the last section of the chapter Arrow Addition) for real numbers a, b and c that c·(a,b) = (ca,cb) without any reference to or use of the add the angles, multiply the lengths arrow multiplication rule. But c = c+i0 = (c,0) gives a point in the plane. So we can multiple c = c+i0 = (c,0) and (a,b) = [r,q] using the add the angles, multiply the lengths rule. Two cases, more precisely possibilities, will be examined.Case 1: c ³ 0 Observe for c > 0 that c = c+i0 = [c,0] has angle 0 degrees and length c = |c|. Thus the add the angles, multiply the lengths multiplication rule yields
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Case 2: c < 0 Now c = -d < 0. But d > 0 implies
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Conclusion. Multiplication of a point (a,b) by a real number c = c+i0 with and without the add the angles, multiple the lengths rule gives (ca,cb).
Some Vocabulary.
For each point or complex number z = a+b i = (a,b) = [r,q] in this plane, we say that a is the real part of z; that b is the imaginary part of z; that r = |z| = Ö[(a2+b2)] is the magnitude, modulus or absolute value of z (different texts prefer different terms); and that q is the angle or argument of z.Remark.
The use of round brackets () in the notation for rectangular coordinates (a,b) stems from the convention in many algebra texts written before this one. The use of square bracket [] in the notation for polar coordinates [r,q] here was chosen simply because the square brackets were available. In retrospect, cosmetic appearance alone would suggest the employment of round-brackets for polar coordinates and square brackets for rectangular coordinates. The development of notation is not always cosmetically optimal.Three Problems.
- Locate in the plane the complex conjugate and reciprocals of the complex three numbers s = 3+4i, t = 12+(-5)i, and z = cos(120°) +isin(120°).
- Locate the three complex cube roots of 1 (unity) .Hint: divide the unit circle into three arcs each spanning an angle of 360/3 =120 degrees. The required roots are at the ends of each arc (if two arcs share the endpoint 1 = 1+i0.
- Locate the fourth, fifth and sixth roots of unity. What is the general pattern for n-th roots of unity (where n = 2, 3, 4, ¼).?
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.