Premath: Properties of Number & Space
I have entitled this section "Pre-Math" because most of the topics we will cover in this course are not "math" as a mathematician might consider it; rather, these topics are the rudiments that are required to do true mathematics, namely proof and less conceptually concrete problems. Many of the topics in this course will be familiar to most who have gone through a public school mathematics curriculum: pre-algebra, algebra (I and II), Euclidean geometry, Trigonometry, and other topics in "pre-calculus." Additional to these traditional topics, we will discuss topics in Number Theory, Combinatorics, Graph Theory, and Game Theory.
For most of the past 150 years, mathematics has been taught with calculus as the seeming apex of understanding, causing all course materials to work towards producing a student extremely proficient in doing calculus. This model has many historical reasons for existing, but it need not be this way and perhaps should not be this way anymore for similar historical reasons. Thus, though this course will certainly prepare one for doing calculus, it will also give them many other mathematical tools, perhaps not useful to calculus, but other modern and more relevant fields of mathematics that the student will study in later years at Aegis Institute.
chapter implicit functions instead of conic sections because I feel that, for the most part, modern textbooks are emphasizing the wrong ideas here. Part of the beauty of studying conic sections is that the geometry of objects are captured in cartesian relationships. Note that I did not use the word functions. Strictly speaking, they are not functions (though, it is true for the precocious amongst you that we can extend our idea of what a function is and parameterize almost anything!).
It turns out, though, that this is generally fine. They imply a collection of functions hence we group them in a very broad category known as implicit functions. The point of studying conic sections is to realize that given a relatively innocuous algebraic object like
we must engage in relatively intensive study to understand all the nuances implied by this equation. If we do this thoughtfully enough, we may recognize certain patterns that lead us to us towards standard forms of these equations (Mathematicians once much preferred the phrase "canonical forms" because it harkens towards a collection of deep theories at the heart of mathematics known as canonical form theorems).
The importance of these standard forms is that, at a glance, I can glean most of the properties of the mathematical object I am looking at without much thought. The importance of algebra is that it is a parsimonious language. It captures infinite objects in all their majesty in tiny, finite collections of symbols.
Broadly speaking there are two very distinct types of conic sections: compact and noncompact. One way to think about compact objects is that I can draw a representation of their entire graph on a single chart.
is our compact conic section with two special cases. Let's
examine the standard form of the ellipse and see how it's related
to the above equation (Note that a and b in the standard form of the
ellipse are not the same a and b from the more general quadratic form
above). The h and k should appear all too familiar from the section on
transformations of functions. They move a special point- in this case,
the center- to a new place in the cartesian plane. The values of a and
b specify the lengths of the major and minor axes and give a way to
determine the locations of the foci. We call 2a the length of the major
axis if a > b and 2b the length of the major axis if b > a.
The implicit functions are given in orange and blue here. I have
suggested a way to parse up the ellipse so each half is can be defined
in terms of familiar functions. It is, of course, possible to parse this up
into more functions. If we are perceptive in analyzing the standard form
of the ellipse and the original equation we will that is necessary, but not
sufficient that both a and c in the original equation are positive. Why?
Let's do a concrete example and then talk about special cases.
The standard form was highlighted in green because it contains more useful information for graphing the ellipse. Since a > b, we see the major axis is horizontally oriented. The center can be found at the coordinate (-1, -2). The vertices are then found by subtracting or adding a and b respectively. One can then find other useful quantities for graphing such as the coordinates of the foci in the following way
Since the foci are always located along the major axis, they have coordinates . As promised, we'll discuss the degenerate cases. If a = b then we have a circle with radius a . We often write this equation in a slightly modified standard form as . If, additionally, r = 0, we say this is a point at (h, k). Finally, for the time-being, we will only consider r as positive or zero and similarly the major and minor axes lengths of an ellipse will be required to be positive. Again, this is not strictly necessary, but we cannot graph the result on a cartesian plane!
A Step Further
Let's explore another relatively obvious question before we continue with noncompact conic sections. Why is there no fxy term in equation 1? Let's explore an ellipse given by the following equation:
This is an example of ellipse that is rotated off axis. In this case, we find the ellipse is at a 45 degree angle to the x-axis but maintains the expected center of (0, 0). It turns out that we need never worry about this situation because of an elementary result that will you see in your linear algebra course. There is always a way to transform the set of points that respects size and shape of the ellipse, but reorients it vertically or horizontally.
Incidentally, you have all the tools you need at your disposal to implicitly solve the equation that gives rise to this ellipse. How would you do it? Including the cross term given by fxy also serves to greatly complicate our understanding of this mathematical object because it allows for the introduction of numerous new degenerate cases (like radius 0 circles!).
Parabolas and Hyperbolas
We'll briefly do an example with a horizontally oriented parabola recalling that we have seen vertically oriented parabolas before. Much like the ellipse or the circle, we have a standard form that is best obtained through completing the square. This has the familiar standard form, but we will add something that will generate a bit more information. Many texts will refer to the following as vertex form, but for our intents and purposes, we will call it the standard form a parabola from this point forward. For our purposes, we will always default to calling the form that allows us to determine what the equation represents with the most information available at first glance the standard form of a geometric object.
The standard form here immediately tells you the vertex of the parabola and the
distance to the focus and the directrix. It also tells you the dilation and the direction
of the parabola.