Chapter 4. Complex Numbers
and Why Slopes
Volume 1B, Mathematics Curriculum Notes
Reason itself appears in mathematics whenever we draw a conclusion from a
calculation, a diagram or from some implications rules employed in
mathematical definitions and assertions. The physicist Richard
Feynman (1918-1988) gave three public lectures at McGill University
in 1979 - printed version says 1976, oops. His work on physics has
been followed by many scientists and students [1].
[1] Apart from his more serious works, he is the author of two light
books, Surely You are Joking, Mr. Feynman! Adventures of a Curious
Character, 1985 and What Do You Care About What Other People
Think? Further Adventures of A Curious Character, 1988, both
distributed by Norton, New York.
In the lectures, most likely tongue-in-cheek, he suggested that physics
was based on two easily described operations, namely the addition and
multiplication of arrows in the plane. His description of arrow
addition and multiplication for a general, non-mathematical
audience was a model for the informal, very visual, yet also
adequate, presentation of mathematical ideas. He gave it under the
guise of describing physics. And he avoided panic among the
mathematically shy by not saying that the arrows, with their addition
and multiplication, represent what pure and applied mathematicians
(since Gauss) regard as the complex numbers.
Two Topics, Easily Described
Talking about arrows, vectors and slopes are further parts of
mathematics besides rule-based reason and the algebraic way of writing
and thinking. The on-paper presentation of these glimpses is somewhat
technical. They are much easier to understand from a live presentation
[2]
[2] What is not immediately understood below could be left for as
subject further inquiry – questions for someone more familiar with late
high school or early college math.
Arrows, Vectors and Complex Numbers
In geometry and other parts of mathematics, arrows are called vectors.
Arrows and vectors have both directions and magnitudes (or lengths).
For instance, the information that a restaurant with excellent food and
coffee is 100 steps due north of your present location defines a length
and direction to a fine dining area. This vectorial information, the
length and direction, is most appealing when you are hungry and you
have sufficient funds in your pocket. This vectorel information
indicates a possible displacement or movement.
Arrows or vectors, that is lengths with a direction for movement, play
a role in navigation. The small print below shows why, and is optional
reading.
On a map, drawing an arrow from a point A to a point B
represents a linear (straight) displacement or movement between them,
the tail point A and the head point B. To show a second
displacement from the head point B to another point C,
draw a second arrow from B to C. This second arrow with
tail at B and head at C represents a second movement.
The two arrows together show a nonlinear movement from
a tail point A of the first arrow to the head point C
of the second arrow. The straight arrow joining the tail point
A of the first arrow to the head point C of the second
is a third arrow. It is called the sum of the first two.
This small print describes the head to tail method for arrow and vector
addition. The two words arrow and vector are interchangeable. While the
latter is more proper, the former will be favoured in this work.
The description of the multiplication of arrows in the coordinate plane
follows in small print - more optional reading. The add the angles,
multiply the lengths rule in this small print implies a simplification
for exposition of both complex numbers and trigonometry.
A coordinate plane has two axes, one is called
horizontal and the other vertical. The intersection of the two axes is
called the origin. If the tail of an arrow is place at the origin, it
forms an angle with the horizontal axis. The size and direction of the
arrow is given, that is determined by its length and this angle.
Feynman's repeated instruction to obtain or compute the product of two
arrows was as follows: add their angles and multiply their lengths to
obtain the angle and length of the arrow which is their product. Note
the absence of units of length here.
In this example, the angle of the product is given by the sum of the
angles of the factors, that is 69.59° = 22.62°+46.97°
while the product of the factor's lengths (1.026)(1.3) = 1.3338 gives
the length of the product.
The rule add the angles, multiply the lengths is the polar
coordinate method for multiplication. A knowledge of the how to
multiply nonnegative real numbers and of how rectangular (Cartesian)
and polar coordinates locate points in the plane is enough to
understand it. So this multiplication could be explained to students
who have yet to master (i) the law of signs for real number
multiplication and/or (ii) the algebraic way of writing and thinking.
Indeed with the identification of the horizontal axis in the coordinate
plane with real numbers, the rule add the angle, multiply the
lengths can be used to imply or confirm the law of signs. This
add the angles, multiply the lengths rule can be explained and
understood before multiplication by negative numbers is explained. The
latter multiplication can even be introduced by this rule. See the
chapters on complex numbers (basic ideas) in the companion book Why
Slopes and More Math for more details.
Remark. In the companion book, the chapters on complex numbers
show in two different ways how the add the angles and multiply the
lengths rule leads to the standard expressions for the rectangular
coordinates, alias real and imaginary parts, of a product. Here two
ways imply (and do not depend on) the angle sum formulas in
trigonometry. Moreover, add the angles and multiply the lengths
rule along with the standard expressions justifies the use of complex
exponential, and so immediately simplifies the treatment of trig in a
manner familiar to students of engineering and physics.
Why Study Slopes - A Calculus Preview
Slopes are defined and employed in mathematics to describe how the
graph of one quantity versus another rises and falls. Slopes measure
the rate of change of one quantity with respect to another: how fast a
first quantity rises, falls or moves as the other quantity in the graph
changes. A road which rises (or falls) three feet for each one hundred
feet travelled horizontally, or three meters for each 100 meters
travelled horizontally, is said to have a grade or slope of three
percent. This slope gives the rate of change of height with respect to
the distance travelled horizontally along the road. A graph of height
above sea level versus distance along a road with this grade would be a
line segment with a slope of three percent. Different sections of a
road could have different slopes. That is, on different portions,
height could change at different rates. Here the slope of the curve can
be estimated by the slope of a straight edge or piece of wood that lies
on it. How is a technical detail presented today in middle high school
algebra courses.
Why slopes are studied can be shown by a minimal-notation tour of
easily-visualized, geometric interpretations of slopes and
slope-related calculations in the context of skiing[3]. Briefly, the ski-based tour is as follows. A skier
can tell from the slope of the ski (that is the slope sign or value),
whether his or her travel is uphill, downhill or horizontal. In going
over a hill top, the ski slope changes from positive to negative as the
skier goes up and then down. Increases and decreases in the slope of
the ski further show the skier how the steepness of the hill changes.
All these observations have applications elsewhere, but the picture of
a skier travelling over a hill y = f(x) gives a first image of them.
Slopes to graphs and the areas under a graph have many interpretations
in mathematics, physics and business. The interpretation depends on the
quantities graphed.
Speed, acceleration, constants of proportionality are all examples of
slopes. Slope or rate of change calculations, and methods for the
reversal of these calculations, all appear in the algebraic reasoning
of business, science, engineering and technology.
In calculus, the mathematical subject mainly concerned with
slope-related calculation, slopes are called derivatives. Slopes
may be computed or measured from formulas or graphs. Rules for
obtaining or deriving slopes from formulas are explained in
calculus courses and are called rules for differentiation. Using the
rules in reverse, an ad hoc process in calculus with two names,
anti-differentiation or integration, leads to well-known and not so
well-known formulas for areas, volumes, perimeters or distances,
weights and masses, etc. The etc. includes many quantities in physics
and engineering, and sometimes business. The rules for obtaining slopes
or derivatives for graphs given by formulas, and the rules for
obtaining formulas from slope information, all depend on the algebraic
way of writing and thinking.
In the companion book, Volume 3, Why Slopes and More Math, the
geometric and physical interpretations show why slope-related
calculations are of interest without saying how to calculate. The
latter is a detail left to calculus courses. The why slopes chapters in
this companion book provide a ski-based tour, a preview or review of
calculus [3] for students of algebra, trig or calculus, itself. The
appendices in this companion book give the proofs normally omitted from
first courses in calculus.
[3] This tour along with the three skills for algebra and forerunners
to the two logic puzzles were first presented in the fall of 1983 to
a class of weak college students at the now-closed St. Croix Campus
of Vanier College in Montreal alongside colleagues who had their
students interest at heart, and who regretted not being able to do
more for those students.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
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Calculus Starter Lessons
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
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Appetite.
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