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  23 Complex #'s  Back ] Home ] Next ]    

Chapter 23
Complex Numbers - Links to Trig

This applet (online only) illustrates the ideas in this chapter.

Further sections of this chapter are given by the web pages . Use the next, previous and up buttons in these links to move between these sections or to return this page.

The addition of complex numbers or points in the plane was given by means of their rectangular coordinates while multiplication was given in terms of polar coordinates. It is still an exercise perhaps in trig, an application of the angle sum formulas, to obtain expressions for the rectangular coordinates of a product in terms of the rectangular coordinates of the factors. Another exercise or alternative, is to justify and then apply the distributive law for complex multiplication over addition. The application bypasses the exercise or link with trig just indicated in favor some algebraic manipulation. The alternatives are given. Which approach to favor may be a matter of taste. This chapter assumes you are familiar with the unit circle definition of sines and cosines and with the addition of vectors.

The  cis or exponential functions

From trigonometry, recall the unit circle definitions of the sine and cosine functions. Let
cis(q) = cos(q)+isin(q) = exp(iq)
of a purely imaginary argument. It is now easy to say how and why the property
cis(A)·cis(B) = cis(A+B)
follows immediately from the above add the angles, multiply the lengths definition of complex multiplication. Hint: both factors have unit lengths.

Applying the Angle Sum Formula

Note the angle sum formulas for cosine and sine met in this section will be proven later. When a+bi = (a,b) = [R,q] and c+id = (c,d) = [r,b] basic trigonometry gives
a
=
R cos(q)
b
=
R sin(q)
c
=
r cos(b)
d
=
r sin(b)

Now the add the angles, multiples the length rule implies the product (a+bi)(c+id) has polar coordinates [Rr,q+b] = (x,y). The rectangular coordinates (x,y) of the product are therefore given by

x
=
Rr cos(q+b)        and    
y
=
Rr sin(q+b)
But the angle sum formula for cosine say
cos(q+b) = cos(q)cos(b)-sin(q)sin(b)
This implies the real part
x
=
Rr cos(q+b)
=
Rr (cos(q)cos(b)-sin(q)sin(b))
=
(R cos(q))(r cos(b))-(R sin(q))(r sin(b))
=
ad-bc
This expresses the real part x of the product x+iy = (x,y) = (a+bi)(c+id) in terms of the real and imaginary parts of the factors, that is their rectangular coordinates. The argument for the imaginary part is similar. It is given next.

The angular sum formula for sine says
sin(q+b) = sin(q)cos(b)+cos(q)sin(b)
This implies the imaginary part
y
=
Rr sin(q+b)
=
Rr (sin(q)cos(b)+cos(q)sin(b))
=
(R sin(q))(r cos(b))+(R cos(q))(r sin(b))
=
bc+ad
This expresses the imaginary part y of the product x+iy = (x,y) = (a+bi)(c+id) in terms of the real and imaginary parts of the factors, that is their rectangular coordinates.

Product Rule in terms of Rectangular Coordinates

The foregoing yields the expression
(a+ib)(c+id) = (ac-bd)+i(bc+ad)
for the product of two points in the plane in terms of their rectangular coordinates, alias real and imaginary parts.

Properties of Complex Numbers

The assumption that the addition and multiplication of positive real numbers a, b, and c are commutative and associative operations implies the following. The length and angle defined multiplication of complex numbers is a commutative and associative operation as well. Details are omitted. They are left as an exercise.

Another Exercise. Employ the expresson (a+ib)(c+id) = (ac-bd)+i(bc+ad) to show that the distributive law for real numbers and the above expressions for the real and imaginary part of the product of two complex numbers implies the distributive property a(b+c) = ab+ac holds for any triplet a = a1+ia2, b = b1+ib2 and c = c1+ic2 of complex numbers.

 

Why Slopes
and
More Math

understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7

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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
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


If  you like Volume 3  you may also like  Three Skills for Algebra , Exponents & Radicals Exactly,  complex numbers, Euclidean Geometry , More Calculus and  Pattern Based Reason  as well. 

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

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.


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