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Chapter 22
Complex Numbers - Basic Ideas
Here is a geometric story which describes the complex numbers, or what
mathematicians since Gauss in the 1840's have regarded as the complex numbers.
This geometric story provides a confirmation of the law of signs for real
numbers. This first part of the story could be explain to someone familiar
with arithmetic and the measurement of coordinates, rectangular and polar in
the plane. The second part of this story is an exercise in trig. The second
part appears in the next chapter.
This applet (online only)
illustrates the ideas in this chapter.
This first half assumes you have some familiarity with the measurement of
distances and angles, with the addition of real numbers and points in the plane,
and finally with multiplication of nonnegative (that is zero and positive) real
numbers.
1. The immediate motivation for this approach (in this chapter) stems
from three successive 1976 McGill University public lectures of the late
Richard Feynman. He simply described physics as the addition and
multiplication of arrows in the plane. He defined their multiplication as
follows: add their angles and multiply their lengths. In terms of the
polar coordinate [r1,q1]
and [r2,q2] for the
factors, the polar coordinates of the product is [r,q]
= [r1r2,q1+q2]
.)
All this was effectively presented to a general audience with no mention of
vectors nor the Gauss-Argand representation of complex numbers. See the
foreword.
2. In Morris Kline's three-volume work Mathematical Thought from
Ancient to Modern Times, see in volume 2, Chapter 27, the third
section called The Geometrical Representation of Complex Numbers. This
section briefly describes the approach of Caspar Wessel (1745-1818). Part of
Wessel's work (translated into English) is reproduced in David Eugene Smith's
1929 work A Source Book in Mathematics, Dover 1959 Reprint.
Points in the Plane
Addition
The sum of two points with the rectangular coordinates (a,b)
and (c,d) is given by (a+c,b+d). We
therefore write
For example (2,5)+(6,2) = (8,7).
In words, the addition rule is simple add the rectangular coordinates of
the summands to get the rectangular coordinates of the sum. With this in
mind, the following question is easy: What are the rectangular coordinates of
the sum of (1,14) and (2,8)? Answer: (1,14)+(2,8) = (1+2,14+8) = (3,22). The
chapter Arrow Addition discusses the addition of points or arrows in
the plane further.
Multiplication
Next we define using polar coordinates the product of two points in the
plane. Each point or factor is located by means of angular displacement or
rotation from the positive real axis, and also a nonnegative distance from the
origin. The product of two points is given by a third point. Its angular
displacement is the sum of the angular displacement of the factors. Its distance
to the origin is the product of the distances of the factors. This is the add
the angles and multiply the lengths rule. In polar coordinate notation, the
multiplication rule and definition is indicated by
| [r1,q1]·[r2,q2]
= [r1r2,q1+q2] |
|
(As in the previous chapter, square brackets are used to indicate polar
coordinates while round brackets indicate rectangular coordinates.)
Example. Two arrows are to be multiplied. One has length 1.3 and angle
22.62°; the other factor has length 1.026
and angle 46.97°; and so their product
has length 1.3338 = 1.3·1.026 and angle 69.69°
= 22.62°+46.97°;
and that is it. See the following diagram.
Another Example. The product of the two points [3,80°]
and [4, 60°] is [(3)(4), 80°+
60°] = [12,140°]
Stop For a Summary
The addition of points in the plane is given by means of their rectangular
coordinates while multiplication is given in terms of polar coordinates. It is
an exercise in trig to obtain expressions for the rectangular coordinates of a
product in terms of the rectangular coordinates of the factors. See the next
chapter. (Note that while your reading of this chapter requires a mastery of the
algebraic way of writing and thinking, you could explain with diagrams and
examples only, the ideas in this chapter to someone without this mastery. It is
the on-paper communication which requires the mastery.)
Introducing the Complex Numbers
Points in the plane with the operations of addition and multiplication just
given are called the complex numbers. The plane with these two operations on its
points is called the complex numbers plane, or more briefly the complex numbers.
We will now change to a more standard notation for them. We may and often
will write the rectangular coordinates z = (a,b) as z
= a+ib, We will further call the abscissa a, the real part
of the complex number z = a+ib. We will also call the
ordinate b, the imaginary part of the complex number z = a+ib.
We will say that the complex number z = a+ib is purely
imaginary when its real part a = 0. The angle of a purely imaginary
complex number z = a+ib = 0+ib = (0,b) is 90
degrees or 270 degrees (modulo
FOOTNOTE: Two quantities x and y are equal modulo a third
quantity c, if and only if their difference x-y
= kc for some whole number or integer k.
360 degrees), depending on the sign of the imaginary part b. When b
> 0, the angle is 90 degrees (modulo 360 degrees). When b < 0, the
angle is 270 degrees (modulo 360 degrees).
We will also say that z = a+ib is (purely) real when its
imaginary part b is zero. The angle of a (purely) real complex number z
= a+ib = a+i0 = (a,0) is 0 degrees or 180
degrees (modulo 360 degrees), depending on the sign of the real part a.
If a > 0, this angle is 0 degrees (modulo 360 degrees) while if a
> 0, this angle is 180 degrees (modulo 360 degrees).
Real Numbers as Complex Numbers
Each complex number z = a+i0 with imaginary part zero
gives and is given by a real number a. We will write z = a
in this situation, and say that the complex number z is also a real number.
With this practice, the real numbers can be regarded as a subset of the
complex numbers; and the real number line can be identified with the horizontal
axis of the plane.
Confirmation of The Law of Signs
We identify the real number line with the horizontal axis of the plane. With
this identification, observe that positive numbers have angular displacement
zero, modulo 360 degrees. Also observe that negative numbers have angular
displacement 180 degrees, modulo 360 degrees. The magnitude of a real number is
its distance to the origin.
Suppose z = a+i0 and w = c+i0. We
want to compute the product zw with the multiply the lengths, add the
angles rule. Each factor has length |a|
or |c|. Each factor
has angle 0 or 180 degrees (modulo 360 degrees). The relationships
0° = 0°+0°
180° = 0°+180°
= 180°+0°
360° = 180°+180°
= 0° (modulo 360°)
imply the add the angles, multiply the lengths rule for the
multiplication of complex numbers agrees with the ordinary method for
multiplying real numbers and the law of signs. The relationship in particular
imply
- (+1) = (+1)(+1) as 0° = 0°+0°
- (-1) = (+1)(-1) = (-1)(+1)
as 180° = 0°+180°
= 180°+0°
- (-1)(-1) = (+1) as
360° = 180°+180°
Examples and then some further comments may reinforce these ideas. For the first
example, the number 4 is now identified with the point (4,0) = [4,0°]
= [4,360°]. This number or point has
distance 4 to the origin and angle of 0°,
modulo 360 degrees, with the horizontal axis:

For the second example, the number -2
is identified with the point (-2,0) = [2,180°].
See the figure below.
Now multiplying the point [2,180°]
by itself leads to the product [2,180°]2
= [22,180°+180°]
= [4,360°] = [4,0°].
Thus the point on the horizontal axis identified with -2
when squared gives the point identified with +4 indicated above. The 360 degrees
in the diagram for the number or point 4 = (4,0) represents the doubling of the
angle 180 degrees.
For an example or exercise, compute the pairwise products of 3=3+0i, 4=4+0i,
-3=-3+i0 and -4=-4+0i using the add the angles, multiply the lengths rule.
Remark. The add the angles, multiply the length rule could be
used to define the product of real numbers to people/students who know (a) about
the addition of real numbers or coordinates and (b) about the multiplication of
non-negative numbers. They would not need to have any previous knowledge of the
law of signs.
FOOTNOTE: The add the angles, multiple the lengths rule for the
multiplication of complex numbers thus yields a rule for the multiplication of
real numbers once the multiplication of positive numbers with themselves or
zero is understood/defined.
More Exercises. Compute the following using the multiply the lengths,
add the angles rule:
- A = (1.5)·(2).
- B = (1.5)·(-2).
- C = (-1.5)·(-2).
- D = (1.5)·(-2).
- E = [10,45°] ·[[1/20],15°].
Note each factor gives a point or arrow in the coordinate plane.
Stop For A Summary. The polar coordinate definition
| [r1,q1]·[r2,q2]
= [r1r2,q1+q2] |
|
of the product of two point in the plane, involves the multiplication of lengths
(= distances to the origin) and the addition of angles. For points on the
horizontal axis, the angles of the factors are zero or 180°
(modulo 360°). Computing the angle of the
product will involve one of the following expressions:
Since the angle 180 degrees is associated with -1, and the angles 0 and 360
degrees are both associated with the number +1, the polar coordinate definition
of multiplication of points in the plane agrees with (or yields) the law of
signs for the multiplication of positive and negative numbers.
Square Root of -1
The real number -1 = -1+0i
= [1,180°] has angle 180 degrees (mod 360
degrees) and length 1. The purely imaginary number (0,1) = 0+i1 = [1,90°]
has angle 90 degrees and length 1. Multiplying this point or number by itself,
that is, squaring it, gives the point with length 1 ×1 = 1 and angle 90°+90°
= 180°. So the product equals -1+0i
= -1. We call i = the principal square root of
-1.
A second square root of -1 is obtained as follows. The imaginary number (0,-1)
= 0+i(-1) = [1,-90°]
has angle -90 degrees and length 1. Multiplying this
point or number by itself, that is squaring it, gives the point with length 1
times 1 =1 and angle (-90°)+(-90°)
= -180° = 180°
(mod 360°). So this product equals -1+0i
= -1 as well.

This provides two square roots of -1 as both [1,+90°]2
= [1,+180°] = -1
and [1,-90°]2
= [1,-180°] = -1.
Square Roots of Other Complex Numbers
The square root of a positive number or zero are real nonnegative numbers. I
assume in the following that you know how to compute these square roots. The
square roots of negative numbers and of other arrows or points in the coordinate
plane depend on this ability.
Observe that squaring points in the plane doubles their angular displacements
and squares their magnitudes (distance to the origin). That is, the add the
angles, multiple the lengths rule gives
| [r½, |
1
2 |
q]·[r½, |
1
2 |
q] = [r
,q] |
|
Therefore the arrow [r½,[1/2]q] when
squared (meaning multiplied by itself) yields [r,q]
. So it is called a square root of the arrow [r,q].
Another square root is located by the polar coordinates [r½,[1/2]q+180°]
since [r,q] = [r,q+360°]
both locate the same point in the plane. You should consider the special case of
positive numbers z = a+i0 = [a,0°]
where the angle q = 0 degrees.
Exercises.
- Find all the square roots of 4 and -4 and plot them.
- Find the cube roots of 27 and -27 and plot them in the plane.
- Find the square roots of \cis(45°)
= cos(45°)+isin(45°)
= [1,45°].
Complex Conjugates
The complex conjugate of a complex number z = a+b i
with polar coordinates r = Ö[(a2+b2)]
and q is the complex number [`(z)]
= a-b i with polar coordinates r
and -q.
* Exercise. Show multiplying a complex number a+b
i by its conjugate a-b i
gives the nonnegative number r2 = a2+b2.

Conjugates and Reciprocals
Observe that p = [(a)/(r2)]-i[(b)/(r2)]
= [1/(r2)][`(z)] has angle -q
and length [1/(r)]. Here p = [1/(r2)][r,-q]
= [[1/(r)],-q].) Multiplying number p =
[[1/(r)],-q] by z = [r,q]
gives the complex number [1,0] with length 1 and angle 0, that is, the real
number 1. And multiplication of any point (c,d) by 1 = [1,0°]
yields back the point (c,d)
The reciprocal (or multiplicative inverse) of the complex number z = a+b
i with length r > 0 and angle q is
the complex number p with length 1/r and angle -q.

Observe that if r > 1 then the length of the reciprocal [1/(r)]
< 1 < r, that is, the length of the reciprocal is less than 1 and
the length of the original number. In contrast, if 0 < r < 1 then
[1/(r)] > 1 > r. Question: Which of these two cases is
represented in the above diagram? What happens in the case r = 1?
Two Algebraic Properties
Observe
| [r1,0]·([1,q]·[r2,q2])
= [r1,q1]·[r2,q2] |
|
since [r1,0]·([1,q]·[r2,q2])
= [r1,0]·[r2,q1+q2]
= [r1r2,q1+q2]
. Similarly
| [1,q]·([r1,0]·[r2,q2])
= [r1,q1]·[r2,q2] |
|
Real Multiples of Arrows
We said earlier (in the last section of the chapter Arrow Addition) for
real numbers a, b and c that c·(a,b)
= (ca,cb) without any reference to or use of the add the
angles, multiply the lengths arrow multiplication rule. But c = c+i0
= (c,0) gives a point in the plane. So we can multiple c = c+i0
= (c,0) and (a,b) = [r,q]
using the add the angles, multiply the lengths rule. Two cases, more
precisely possibilities, will be examined.
Case 1: c ³ 0 Observe for c
> 0 that c = c+i0 = [c,0] has angle 0 degrees and
length c = |c|.
Thus the add the angles, multiply the lengths multiplication rule
yields
| c·(a,b) = [c,0]·[r,q]
= [cr,0+q] = [cr,q]
= (ca,cb) |
|
as before.
Case 2: c < 0 Now c = -d
< 0. But d > 0 implies
| (d,0)·(a,b) =
[d,0]·[r,q] = [dr,q]
= (da,db) |
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Therefore
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| (-1)·[d,0]·(a,b)
= (-1)·[dr,q] |
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again.
Conclusion. Multiplication of a point (a,b) by a real
number c = c+i0 with and without the add the angles,
multiple the lengths rule gives (ca,cb).
Some Vocabulary.
For each point or complex number z = a+b i = (a,b)
= [r,q] in this plane, we say that a is
the real part of z; that b is the imaginary part of z;
that r = |z|
= Ö[(a2+b2)] is
the magnitude, modulus or absolute value of z (different
texts prefer different terms); and that q is the
angle or argument of z.
Remark. The use of round brackets () in the notation for
rectangular coordinates (a,b) stems from the convention
in many algebra texts written before this one. The use of square bracket [] in
the notation for polar coordinates [r,q]
here was chosen simply because the square brackets were available. In
retrospect, cosmetic appearance alone would suggest the employment of
round-brackets for polar coordinates and square brackets for rectangular
coordinates. The development of notation is not always cosmetically optimal.
Three Problems.
- Locate in the plane the complex conjugate and reciprocals of the complex
three numbers s = 3+4i, t = 12+(-5)i,
and z = cos(120°) +isin(120°).
- Locate the three complex cube roots of 1 (unity) .Hint:
divide the unit circle into three arcs each spanning an angle of 360/3
=120 degrees. The required roots are at the ends of each arc (if two
arcs share the endpoint 1 = 1+i0.
- Locate the fourth, fifth and sixth roots of unity. What is the
general pattern for n-th roots of unity (where n = 2, 3, 4, ¼).?
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Why Slopes
and
More Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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Read slowly, Volumes 2 & 3 may ease or avoid
calculus difficulties. Take the risk.
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Chapters and Appendices
Content Guide Foreword 2nd Content Guide 1. Introduction Geometric Calculus Preview (1983) 2. Algebraic 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
Appendices:
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.
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
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For Parents & Teachers: Speaking
Skills, Reading
& Writing,
Preparing for Science, ends,
values and methods for work and study, parent- friendly mathematics booklets for ages 4-14.
-
Math
Education
Essays (opinions,
possibilities, references)
- POMME, a two
level program for future skill development in
schools and colleges worldwide. Address content &
motivation gaps with ends, values & methods for skill
development to say which way to go, how and why. -
Present Day Curriculum:
(A) Secondary
I Mathematics
consolidate fractions and measurement, skills and
sense consolidation,
(B)
Secondary II Mathematics
year of algebra and proportionality
(C) See too the following:
- Arithmetic
& Number Theory Practices (horribly put, but
useful)
- Algebra and
Logic SubProgram
(well put, extremely useful)
For Avid Readers in School & Out -
Online Books
1. Elements of
Reason. 1996
1A. Pattern
Based Reason 1995
1B. Math
Curriculum Notes 1996
2. Three
Skills for Algebra 1995
3.Why
Slopes & More.Math
1995
Tour their forewords.
Calculus Prep or Help: See Volumes 2 & 3,
and this bigger
Calculus
Guide.
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Senior
High School &
Calculus Students
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Free Live Lesson
- Operations with Decimals - Comparison, Subtraction and Long Division
- Click here
to attend.
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For Senior
High School Mathematics & Calculus
Intro to Solving
Linear Equations
- a different paths for junior and even senior high
school students.
5
wordy Logic
Chapters
4 curious Algebra
Chapters
Words before & besides symbols. A Key Algebra
forward & backwards Chapter
First Calculus
Preview (1st intro)
Four Calculus
Chapters
(2nd intro)
Intro to Complex
Numbers (long)
Intro to Mathematical
Induction (romantic & wordy at first)
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Many More Topics
1. Decimal
Arithmetic Reference!
2. Integers
- Intro to Signed No.s
3. Fractions
- fully explained.
4. Fractions
with Units
5. Number
Theory,
6. Solving
Linear Equations
7 Formulas
Use Forward & backwards -
8. Proportionality,
Back- & For-wards.
9. Logic
Chapters:
10. Euclidean-Geometry
11. Slopes
& Equations of Straight Lines. (Take
I. See take II below)
12. Why
Study Slopes.
13. Maps,
Plans, Similarity & Trig,
(Take II included here)
14. Quadratics:
Starter lessons
15. Polynomials:
Starter lessons
16 Why
Factor Polynomials:
17 Functions
- Forwards & Backwards.
18. Exponents,
Radicals & logs.
19. Complex
Numbers before trig (new advance/ starter lesson)
20. DC
Electric
Circuits Etc
21. Real
Analysis
22. The
Olde Complex No, Trig
& Vector Section.
23. More
Calculus Stuff
- written after Volumes 2 and 3.
More For Instructors
-
Education
Essays
(opinions,
possibilities, references)
- POMME, a two
level program for instruction K1-14
- Math & Logic How-TOs
1. Arithmetic
2. Algebra
3. More Algebra
4. Beginner Geometry
5. More Geometry
6. Calculus
7. Show Work or Logic
These may be too dense for students.
Level I Material: New Stuff
Time and Date Matters
Level I Arithmetic.
Money Matters
Measurement Matters
Matters of Chance (Risk Control)
Logic
Chapters
(leave what's not clear in Level I to Level II)
Using/Making Maps and Plans.
(A variant of
Maps,
Plans, Similarity & Trig, to
appear here).
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Skill Development Tips
For All
Standards: (A) Take
care to avoid the domino effect of errors & approximations; (B) Do and
record steps in an manner that allows skill mastery to be seen or
corrected. Anything represent substandard work.
Key Numerical Methods
- To multiply signed numbers, prefix the product of their signs to the product
of their lengths or unsigned parts. The product is negative if the no of
negative sign in it is odd.
- To add signed numbers with like signs, prefix the common sign to the sum of the
lengths.
- To add signed numbers with opposite signs, prefix the sign of the longest to
the difference: length of longest minus length of shortest.
- Should we study roots and powers of real numbers with formulas involving exponential and log.
- How does adding and multiplying points in the plane and rotating the midpoint
of a line segment lead to mastery of complex numbers and the thought-based
development of their properties, all before trig?
- New Axioms for High School Mathematics: In accounting, totals of assets
and debts may be calculated by dividing the assets and debts into
non-overlapping (disjoint) groups and then adding subtotals. In general, sums
(and products) of counts and numbers, positive and negative numbers
included, can be obtained by adding subtotals (and multiplying
subproducts, respectively). These practices may be cast as axioms in
secondary mathematics. Then operations on polynomials are easily implied
justified by these "axioms" and the geometric introduction of column
methods for expanding a products of two sums. While set theory in pure
mathematics may imply the above axioms in university mathematics programs
instruction, an earlier and more accessible explanation based on easily accepted
and understood geometric and counting practices derivation of the above
axioms is possible at the high school for students heading for college programs
in science.
In Volume 2: Prep for Calculus
- What is the difference between saying A if B and saying A if and
only if B. Being aware of the difference will sharpen ye wits.
- What is a chain of reason?
-Are your arithmetic skills OK?
-Have words been missing in the introduction of algebra?
- Can ye talk about numbers & quantities varying apart from or before the
use of letters & functions?
- Do ye know about the forward & backward use of formulas?
-Contrapositive: is that backward use of A if B?
-What is a variable x? Answer before speaking of function f(x) = x.
-What a twist! There are no rules of algebra for subtraction and division. But
if you replace them by addition of -x and multiplication by 1/x, rules of
algebra (properties of arithmetic) can be used.
In Volume 3: Calculus Slowly?
-Why are slopes studied and polynomials factored in high school?
- Volume 3 suggests how to ease or delay algebra shock in
calculus *& beyond. In Calculus, derivatives and integrals introduced and defined by limits, but calculated
without when possible by using differentiation rules forwards and backwards. The
second site calculus section may help in differential calculus.
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