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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
Learn to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes your attention. |
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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 23 Complex Numbers - Links to Trig Continued
Before reading the second proof below, explore the geometry-vectors-trig
section of this website created after this book was written. The latter
section presents a simpler path for the exposition geometry, vectors and trig.
The second proof below was part of the exploration of ideas that led to the
simpler path.
Second Proof
Step 1. Suppose P = a+i b and Q = c+id are at
unit distance from the origin of the plane. Then
a2+b2 = 1 and c2+d2 = 1. (Units of length are
omitted. The product P·Q has length 1.) The diagram
below shows that the product P·Q located by adding
angles (and multiplying lengths) coincides with the vector given by the sum of
a·Q and i b·Q. This implies the distributive
law (a+ib)·z = a·z+ib·z for the situation
depicted. The argument holds regardless of the quadrants in
which P is located.
The product a·Q is collinear with the vector Q as
a is real while the product ib·Q is perpendicular
to Q since ib has angle 90 degrees with the positive
(real) axis.
Step 2. The foregoing implies the distributive law (a+ib)·z = a·z+ ib·z for case a2+b2 = 1 = |z|. Since the polar
coordinate defined product of points in the plane is
commutative, the foregoing law is two-sided. Hence z = c+id
with c and d both real and c2+d2 = 1 implies
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a·c+a·id + b·ic+ib·id by step 1 twice |
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due to the definition of the product of complex numbers
and the associativity (proof?) of the addition of points in the
plane. The real part of the product is ac-bd and the
imaginary part is ad+bc.
Step 3. From the polar coordinate definition of the product of complex
numbers, we observe
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[cos(q)+isin(q)][cos(b)+isin(b)] = cos(q+b)+isin(q+b). |
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Now the angle sum formulas for sine
and cosine:
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cos(q)cos(b)-sin(q)sin(b) = cos(q+b) |
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and
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cos(q)sin(b)+sin(q)cos(b) = sin(q+b) |
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follow from the expressions derived above for the
rectangular coordinates of the product of two unit magnitude
points in the plane. How to compute the real
and imaginary parts in a product of any pair of complex
numbers in terms of the real and imaginary parts of the factors is given next.
Distributive Law and Consequences
The next pages offer another way to obtain the
distributive law for multiplication without assuming the
factors have unit length. The next pages further imply the formulas
for the real and imaginary parts of a product, and from them the
angle sum formulas. This shows links between complex
numbers and trigonometry can be obtained in different ways.
The previous sections show how multiplying points (a,b) in the plane
with polar coordinate-based add
the angle, multiple the lengths rule, led to the
expression for the product in terms of the rectangular
components, that is, the real and imaginary parts of the factors.
This section provides an alternate approach. The
expressions are consequences of the distributive law for
complex multiplication.
The Commutative Law
Since the order of multiplication of positive numbers, and
the order of addition of real numbers is immaterial, the
add the angles, multiply the lengths rule implies
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[r1,q1]·[r2,q2] = [r1r2,q1+q2] = [r2r1,q2+q1] |
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Therefore
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[r1,q1]·[r2,q2] = [r2,q2]·[r1,q1] |
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This says that the order of multiplication is not
important. Therefore, the right and left distributive laws
imply each other (why?).
So if one holds, then so does the others.
Use of The Distributive Laws
For every triple of complex
numbers (A,B,C), the right distributive law (the right form)
and the left distributive law (the left form)
applied twice in succession would imply the following in terms of
rectangular coordinates if they held.
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if the right distributive law holds, |
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{(a,0)+(0,b)}·(c,0)+{(a,0)+(0,b)}·(0,d) |
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{(a,0)·(c,0)+(0,b)·(c,0)} |
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+{(a,0)·(0,d)+(0,b)·(0,d)} |
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if the left distributive law holds, |
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+{(0,ad)+ b[1,90°]·d[1,90°]} |
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{(ac,0)+(0,bc)}+{(0,ad)+ bd[1,180°]} |
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{(ac,0)+(0,bc)}+{(0,ad)+ bd(-1,0)]} |
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{(ac,0)+(0,bc)}+{(0,ad)+ (-bd,0)} |
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In terms of complex number notation, the foregoing says
that
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(a+ib)·(c+id) = (ac-bd)+(bc+ad)i |
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where i = Ö(-1).
Therefore
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(a+ib)·(c+id) = (ac-bd)+(bc+ad)i |
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holds for all real numbers a, b, c and d IF the
left and right distributive laws hold.
Now
The property
cis(a)·cis(b) = cis(a+b) follows from the add the
angles, multiply the lengths definition of complex
multiplication and not both factors have unit lengths.
But this property cis(a)·cis(b) = cis(a+b) can be rewritten in terms of rectangular
coordinates or complex number notation as
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(cos(a),sin(a))·(cos(b),sin(b)) = (cos(a+b),sin(a+b)) |
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Computation of the real and imaginary parts of the left
hand side implies the angle-sum formulas
for the cosine function
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cos(a)cos(b)-sin(a)sin(b) = cos(a+b) |
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and for the sine function
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cos(a)sin(b)-sin(a)cos(b) = sin(a+b) |
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respectively.
Proof of Distributive Laws
Plan.
The proof of the distributive law A(P+Q) = AP+AQ will be
based on the observation (the physical assumption) that
multiplication by
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A = [r,q] = [r,0]·[1,q] = [1,q]·[r,0] |
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can be done into two steps. One step
is a rotation through the angle q while the
other is a multiplication by the stretch factor or
shrinkage factor r = [r,0]. Multiplication by a stretch
factor and rotation through an angle will be shown to be
distributive operations over addition.
Distributive Law For Stretch Factors.
Now let P = (a,b) and Q = (c,d).
Now
Therefore
A(P+Q) = AP+AQ when A = (r,0) for some r > 0. This argument
assumes the distributive law for multiplication of the
sum of two real numbers by another.
It can be illustrated
by tiling the plane with parallelograms - copies of the
parallelogram determined by the arrows [1/(n)] P and
[1/(n)]Q (where n ³ 1 is a whole number). Such an
illustration might be sufficient corroboration for some
pre-algebraic students.
Distributive Law for Rotations.
A parallelogram corresponding to the map addition of the
arrows associated with P = (a,b) and Q = (c,d) is indicated below.
We assume that the parallelogram and the
two triangle forming it are rigid bodies. This implies
that after a rotation, that the map addition of the vectors
forming the sides before and after rotation will yield the diagonal arrow
before and after rotation, respectively. See the next diagram.
The following diagram shows that the triangle vertices P, Q and
P+Q rotated respectively into
the triangle vertices P¢, Q¢ and
P¢+Q¢.
This suggests that
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[1,q]·(P+Q) = P¢+Q¢ = [1,q]·P+[1,q]·Q |
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and hence that multiplication by the factor [1,q] is
distributes over the addition of arrows.
End of the Proof. Observe that
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[r,0]·([1,q]·P)+[r,0]·([1,q]·Q |
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([r,0]·[1,q])·P +([r,0]·[1,q])·Q |
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Remark.
The formal or proper presentation of
mathematics relies on no diagrams and on no physical
interpretation or reasoning. The preceding presentation of
complex numbers was informal. It relied on geometric ideas
(assumptions) to make a link between polar and rectangular
coordinates. But the conclusions drawn above can be
obtained in a geometric-free manner (no diagrams) and drawn
solely from assumptions about arithmetic. See the
diagram-free description of the complex numbers and trig
functions in the university-level book
Principles of Mathematical Analysis by
W. Rudin, McGraw-Hill 1964, for more details. | |
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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