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.
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