Chapter 23
Complex Numbers - Links to Trig
This applet (online only) illustrates the ideas
in this chapter.
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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) |
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of a purely imaginary argument. It is now easy to say how and why the property
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
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
But the angle sum formula for cosine say
| cos(q+b)
= cos(q)cos(b)-sin(q)sin(b) |
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This implies the real part
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| Rr (cos(q)cos(b)-sin(q)sin(b)) |
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| (R cos(q))(r
cos(b))-(R
sin(q))(r sin(b)) |
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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) |
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This implies the imaginary part
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| Rr (sin(q)cos(b)+cos(q)sin(b)) |
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| (R sin(q))(r
cos(b))+(R cos(q))(r
sin(b)) |
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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) |
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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.
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Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
|
[ Back ] [ Home ] [ Next ]
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
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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