This was originally going to be the first part of my new conic section notes, but on reflection it’s a little difficult for Y13 students. Thus, because I do think it is interesting, I post it here. (See also, the previous post.)
Quadratic Equations (Again)
You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine. — Hilbert (on an ex-student)
We will study in this topic precisely equations of the form
This class of equations is the class of quadratic equations (in two variables). Note that, without loss of generality, we can assume (otherwise, we just divide through by ).
The set of all points satisfying a quadratic equation is called a curve of the second degree.
Our first theorem is a generalisation of the statement from algebra that every quadratic equation in one variable (i.e. every equation ) has precisely two solutions (if you count correctly).
Theorem. Every line meets a curve of the second degree in precisely two points.
This theorem is very nice, but (like the proof for quadratics in one variable) we do have to make a trade: we have to take our coordinate system to be over the complex numbers. Thus, for the remainder of these notes, we will assume implicitly that rather than living in the real plane we are in fact living in the complex plane . In some sense, then, we are doing geometry in four dimensions. On the other hand, in many places we will only be interested in real coordinates. For example, the graph of a four-dimensional object is a little tricky to draw if we only have three dimensions available!
Our first proof attempt looks all right, and indeed it is almost correct;
Proof: Let be our line. At least one of or is non-zero; assume that is non-zero (but essentially the same proof will work if but is non-zero). Then we can substitute into our curve of the second degree to obtain a quadratic equation in the variable ; by the fundamental theorem of algebra, this equation has two solutions; and each of these corresponds with a point of intersection.
The problem with this proof is illustrated by the following example:- Consider the quadratic . This describes a perfectly good curve in two dimensions, graphed here.
If we intersect this curve with the line , then we obtain a linear equation instead of a quadratic equation — and so even counting multiplicities and moving up to , we only have one solution.
The reader might be tempted to discount this example as uninteresting, because it’s simply the intersection of a pair of two lines with a third line and thus isn’t really a problem that affects our programme of studying curves of the second degree.
The next example, though, is both more fundamental and more worrying. If we intersect the parabola with the line , we obtain a single point of intersection, even up to multiplicity.
Does this mean that we have to abandon the beautiful intersection theorem?
It turns out that, like how we expanded our number system to make the fundamental theorem of algebra work nicely, we can expand our plane further to make our intersection theorem work.
We will not make this precise, but the method we will use is some kind of limiting process; the result, and the system we will work in from now on, is the projective plane over . Let us see how we climb to it.
It turns out that in order to approach some kind of resolution, we need to mention the elephant in the room: the foundation of all the geometry which we have been doing thus far in our mathematical lives. (This might seem at first glance completely unrelated to the algebraic problem we’re trying to solve, but bear with me.)
In ancient Greece, in about 300~BCE, Euclid of Alexandria (in Egypt) wrote his book Elements, one of the most influential mathematical texts of all time. In it, he set down the set of basic results which can be proved true about circles and lines on a plane: while modern geometry has gone much further than Euclid (in considering geometry in more than two dimensions, and on surfaces much more complicated than the plane), the basic results which are taught in any introductory geometry course are usually treated in the Elements.
Euclid was influential because he was the first author (or, more likely, the first author whose work survives) to attempt to rigorously prove all his results based on a set of `universal truths’: a set of simple statements (called axioms) whose truth is self-evident. In some sense, the goal of mathematics is to deduce the behaviour of the most complicated systems possible from the simplest axioms possible. (The axiom system which most of modern mathematics is done within is called ZFC (Zermelo-Fraenkel with Choice, named after two mathematicians working on the area in the early 1900s).) Euclid’s axioms were as follows:
- A straight line segment can be drawn joining any two points.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- Given any straight line and a point not on it, there exists one and only one straight line which passes through that point and never intersects the first line, no matter how far they are extended.
Putting aside the fact that these axioms are in fact insufficient for the purpose of constructing plane geometry, it is clear that the fifth axiom is both far less elegant than the others and far more complicated. Indeed, Euclid himself avoided using it in his proofs except where necessary (he avoids it completely for the first 28 propositions, for example) and for many years mathematicians believed that it was in fact a theorem: that it could be proved from the previous four axioms.
It turns out that this is false: there exist geometries for which the fifth axiom is false (for example, on a sphere).
The mathematical world which we are interested in is known as projective geometry (the name will only make sense towards the end of these notes). We obtain this geometry by replacing the final axiom with the following axiom: Any two distinct lines in a projective geometry meet at exactly one point. This is clearly not true in usual Euclidean geometry. On the other hand, we didn’t let this kind of thing stop us in the past: the number is absolutely useless for counting sheep, for example, but that doesn’t stop the complex numbers from being an interesting (and useful) mathematical space.
The complex numbers became much clearer once we stopped thinking about numbers as a line and started drawing them as a plane. There is an analogous simple way to construct and think about a projective plane by extending the Euclidean plane (and then extending our setting, the complex plane, will be done in the same way).
When we think about it, we find that the only reason that Euclidean geometry doesn’t satisfy this axiom is that parallel lines exist. We will therefore define them not to exist!
More precisely, given every set of parallel lines with slope , we will take our plane and add on a new point called a point at infinity, and then define this new point to be on all the lines with slope . Thus, these parallel lines now intersect at exactly one point (the point at infinity related to their slope) and all other lines intersect in exactly one point (the same place they did before).
This can be made to make some sense physically, and indeed the first place which projective geometry appeared was when Renaissance artists like Leonardo da Vinci began to experiment with perspective drawing — if you look down two parallel lines in real life, they appear to meet at the horizon, and lines going in the same direction away from you (i.e. lines with the same slope) meet at the same point. We call the line joining all the points at infinity (in this physical case, the horizon) the line at infinity.
Going back to our problem of curves of the second degree, let us consider our second example: only intersects the line at one point.
But notice, as the curve tends towards being vertical! More precisely, we have that ; thus, the parabola is eventually parallel to the vertical line, and so the two intersect for a second time at the point at infinity .
This is even more blatant in our first example: we had the following curve (and again, the blue line is the line we intersect with our second degree curve):
In this case, the curve is not only eventually parallel, but in fact it is actually parallel!
The picture to keep in mind, for conic sections at least, is the following.
Note that this picture does not actually reflect what is really going on when we graph second degree equations in the plane — it’s a diagrammatic representation of the fact that we’ve made a fundamental extension to our system of geometry (and to our system of algebra, but this isn’t visible on a real graph).
Philosophically, all we’ve done is decided what property we want our algebro-geometric system to have (that curves of the second degree intersect any line at exactly two places, unless it’s tangent to the line) and then invented a geometry in which this property is true. It turns out that this geometry has a lot of other beautiful properties, most of which we don’t have time to look at — but at this stage, it’s completely synthetic. We simply arbitrarily added a bunch of points (and we’re honestly lucky we didn’t get any contradictions). In fact, eventually we will see that there is an incredibly natural way to construct a projective plane without having to worry about sticking in a bunch more points and cooking it ourselves. For now, we will content ourselves with proving our motivational theorem.
Proof of Theorem
Let be our line. Then we can substitute into our curve of the second degree to obtain an equation in the variable which is either of degree 1 or degree 2. (If necessary, let be infinite.)
Case I. If the equation is of degree two, then by the fundamental theorem of algebra it has two solutions; and each of these corresponds with a point of intersection.
Case II. We will be done if we can show the following: if a line intersects some quadratic equation in exactly one point (i.e. in such a way as to produce a linear equation in rather than a quadratic) then
Suppose does not tend to a finite number. If goes to infinity, then in particular it intersects every non-vertical line at two points (draw a picture), so if the line intersects at precisely one point then it is vertical; thus if then . On the other hand, is either eventually positive or eventually negative (just by inspecting the expression for it above), so it either tends to infinity or to some finite limit.
We will now calculate the limit which it tends to, if it tends to some finite value.
Step 1. We will find the condition on the coefficients of our quadratic for to vanish upon substitution. So let us substitute into our quadratic; we obtain .
Expanding, and setting the coefficient of the term to zero, we find that .
Step 2. We will find the derivative ; this is a simple exercise in implicit differentiation and we get that . Substituting our condition on the coefficients, we have .
Step 3. Taking limits of both sides, we have .
It is not immediately obvious how we should deal with this right-hand side, because also depends on . Looking at the equation for the quadratic, we note that as , we must have — otherwise the right hand side couldn’t stay at zero. Therefore, both the top and the bottom of the fraction under our limit go to infinity; we can apply the following theorem (which we will not prove here):
Theorem. If and , and if , then .
Proof: see Spivak, chapter 11, theorem 9 and exercises 36-38.
Let . Then we have
Thus we have shown that if the derivative tends to any value, it must tend to .