Chapter 20
Vectors and Complex Numbers
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.
Elements of this chapter appear in the simplified
treatment of complex numbers (& trig), posted online at
this site after the completion of Volume 3.
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.
|
About the Next Chapters
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.
|
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
|
|