7 Complex Numbers
Appetizer - A Complex Number Applet
1 Rectangular Polar Coordinates Review
2 Complex Numbers -made easier we hope
3 Addition Properties
4 Multiplication Properties
5 An Easy Proof of the Distributive Law
6 Field Properties of Complex Number
7 Second Way to Calculate Products
8 Unit Circle Development of Trigonometry
9 The complex-number valued trig function cis
10 sine-cosine Angle Sum Formulas via cis
11 sine and cosine double triple angle formulas
12 cis formulas for sine cosines and tangent
13 Trig Formulas for dot- and cross-Products
14 Law of cosines
15 Pythagorean Theorem Converse
16 References and Originality Question
17 Cube Roots of unity
18 Sixth Roots of Unity
19 N-th Roots of Unity
20 N-th Roots of Complex Numbers
21 Logarithms Powers and Exponentials
This folder Complex Numbers Made Easy reproduces and continues the complex
number appetizer or starter lesson in the section above on Cartesian
and Polar Coordinates. It adds easy rotate a midpoint proof of the
distributive law for complex numbers. All other algebraically
described, field properties of complex numbers are consequences of
the algebraically described, field properties of real numbers, alone
or in combination. The site development of the mid-point formula
depends on the Pythagorean theorem.
Site Volume 3, Why Slopes and More Mathematics, includes a proof of
the distributive law that depends on similarity properties and not
the Pythagorean theorem. Whence two ways to calculate the product
of a complex number a + ib of modulus r with its complex
conjugate a - ib implies r2 = a2 +
b2 gives another proof of the Pythagorean. That being
said, the latter proof with the easy midpoint based development of
the distributive law, becomes a confirmation.
The development of complex numbers before or besides the introduction
of unit-circle definition of circular or periodic trigonometric
functions permits the use of complex number properties and techniques
in the derivation and justification of trigonometric formulas.
Lessons provides examaples in the form of trignometric angle-sum,
double-angle and triple angle formulas. Further more, trigonometric
formulas for the dot- and cross-product expressions that appear in
two dimensional vector analysis, the coordinate development, follow
easily from complex number considerations. The cosine law for scalene
triangles, one vertex at the origin, is implied by the trigonometric
formula for dot products in the plane. A converse to the Pythagorean
is an immediate and easy consequence of the cosine law. See lessons 8
to 15.
Cube, sixth and N-roots of unity and N-roots of complex numbers are
described in lessons 17, 18 and 19. The presentation consists of
hand-written, poorly recorded on a pen tablet apart from a computer,
and then uploaded. So the writing and presentation is not optimal.
None the less, teachers and tutors may recognize the essential ideas
and present them in class. The discusion of N-th roots of unity
connects complex numbers to regular N-gons in the plane. Complex- and
real-number for calculating N-th roots are compared and contrasted in
lesson 20.
The discussion of logarithms, powers and exponentials in lesson 21, a
reproduction of the last chapter in site Volume 3, Why Slopes and
More Mathematics, gives a formula based approach that extends the
definition of - the description of how to compute - logarithms,
powers and exponentials from the case of real numbers to case of
complex numbers. This discussion is too deep for high school studies.
It may however serve as appetizer for undergraduate courses in
advanced mathematics.
|
|
For home-tutoring or -schooling, or for schools or colleges
with course content control: Secondary
Mathematics for Ages 11+, A Practical Approach.
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.
|
|