## My 2018 Mathematics A To Z: Quadratic Equation

I have another topic today suggested by Dina Yagodich. I’ve mentioned before her YouTube channel. It’s got a variety of educational videos you might enjoy. Give it a try.

I’m planning this week to open up the end of the alphabet — and the year — to topic suggestions. So there’s no need to panic about that.

The Quadratic Equation is the tool humanity used to discover mathematics. Yes, I exaggerate a bit. But it touches a stunning array of important things. It is most noteworthy because of the time I impressed by several-levels-removed boss at the summer job I had while an undergraduate. He had been stumped by a data-optimization problem for weeks. I noticed it was just a quadratic equation, that’s easy to solve. He was, must be said, overly impressed. I would go on to grad school where I was once stymied for a week because I couldn’t find the derivative of $e^t$ correctly. It is, correctly, $e^t$. So I have sympathy for my remote supervisor.

We normally write the Quadratic Equation in one of two forms:

$ax^2 + bx + c = 0$

$a_0 + a_1 x + a_2 x^2 = 0$

The first form is great when you are first learning about polynomials, and parabolas. And you’re content to something raised to the second power. The second form is great when you are learning advanced stuff about polynomials. Then you start wanting to know things true about polynomials that go up to arbitrarily high powers. And we always want to know about polynomials. The subscripts under $a_j$ mean we can’t run out of letters to be coefficients. Setting the subscripts and powers to keep increasing lets us write this out neatly.

We don’t have to use x. We never do. But we mostly use x. Maybe t, if we’re writing an equation that describes something changing with time. Maybe z, if we want to emphasize how complex-valued numbers might enter into things. The name of the independent variable doesn’t matter. But stick to the obvious choices. If you’re going to make the variable ‘f’ you better have a good reason.

The equation is very old. We have ancient Babylonian clay tablets which describe it. Well, not the quadratic equation as we write it. The oldest problems put it as finding numbers that simultaneously solve two equations, one of them a sum and one of them a product. Changing one equation into two is a venerable mathematical process. It often makes problems simpler. We do this all the time in Ordinary Differential Equations. I doubt there is a direct connection between Ordinary Differential Equations and this alternate form of the Quadratic Equation. But it is a reminder that the ways we express mathematical problems are our conventions. We can rewrite problems to make our lives easier, to make answers clearer. We should look for chances to do that.

It weaves into everything. Some things seem obvious. Suppose the coefficients — a, b, and c; or $a_0, a_1, a_2$ if you’d rather — are all real-valued numbers. Then the quadratic equation has to hav two solutions. There can be two real-valued solutions. There can be one real-valued solution, counted twice for reasons that make sense but are too much a digression for me to justify here. There can be two complex-valued solutions. We can infer the usefulness of imaginary and complex-valued numbers by finding solutions to the quadratic equation.

(The quadratic equation is a great introduction complex-valued numbers. It’s not how mathematicians came to them. Complex-valued numbers looked like obvious nonsense. They corresponded to there being no real-valued answers. A formula that gives obvious nonsense when there’s no answer is great. It’s formulas that give subtle nonsense when there’s no answer that are dangerous. But similar-in-design formulas for cubic and quartic polynomials could use complex-valued numbers in intermediate steps. Plunging ahead as though these complex-valued numbers were proper would get to the real-valued answers. This made the argument that complex-valued numbers should be taken seriously.)

We learn useful things right away from trying to solve it. We teach students to “complete the square” as a first approach to solving it. Completing the square is not that useful by itself: a few pages later in the textbook we get to the quadratic formula and that has every quadratic equation solved. Just plug numbers into the formula. But completing the square teaches something more useful than just how to solve an equation. It’s a method in which we solve a problem by saying, you know, this would be easy to solve if only it were different. And then thinking how to change it into a different-looking problem with the same solutions. This is brilliant work. A mathematician is imagined to have all sorts of brilliant ideas on how to solve problems. Closer to to the truth is that she’s learned all sorts of brilliant ways to make a problem more like one she already knows how to solve. (This is the nugget of truth which makes one genre of mathematical jokes. These jokes have the punch line, “the mathematician declares, `this is a problem already solved’ and goes back to sleep.”)

Stare at the solutions of the quadratic equation. You will find patterns. Suppose the coefficients are all real numbers. Then there are some numbers that can be solutions: 0, 1, square root of 15, -3.5, these can all turn up. There are some numbers that can’t be. π. e. The tangent of 2. It’s not just a division between rational and irrational numbers. There are different kinds of irrational numbers. This — alongside looking at other polynomials — leads us to transcendental numbers.

Keep staring at the two solutions of the quadratic equation. You’ll notice the sum of the solutions is $-\frac{b}{a}$. You’ll notice the product of the two solutions is $\frac{c}{a}$. You’ll glance back at those ancient Babylonian tablets. This seems interesting, but little more than that. It’s a lead, though. Similar formulas exist for the sum of the solutions for a cubic, for a quartic, for other polynomials. Also for the sum of products of pairs of these solutions. Or the sum of products of triplets of these solutions. Or the product of all these solutions. These are known as Vieta’s Formulas, after the 16th-century mathematician François Viète. (This by way of his Latinized, academic’sona, name, Franciscus Vieta.) This gives us a way to rewrite the original polynomial as a set of polynomials in several variables. What’s interesting is the set of polynomials have symmetries. They all look like, oh, “xy + yz + zx”. No one variable gets used in a way distinguishable from the others.

This leads us to group theory. The coefficients start out in a ring. The quotients from these Vieta’s Formulas give us an “extension” of the ring. An extension is roughly what the common use of the word suggests. It takes the ring and builds from it a bigger thing that satisfies some nice interesting rules. And it leads us to surprises. The ancient Greeks had several challenges to be done with only straightedge and compass. One was to make a cube double the volume of a given cube. It’s impossible to do, with these tools. (Even ignoring the question of what we would draw on.) Another was to trisect any arbitrary angle; it turns out, there are angles it’s just impossible. The group theory derived, in part, from this tells us why. One more impossibility: drawing a square that has exactly the same area as a given circle.

But there are possible things still. Step back from the quadratic equation, that $ax^2 + bx + c = 0$ bit. Make a function, instead, something that matches numbers (real, complex, what have you) to numbers (the same). Its rule: any x in the domain matches to the number $f(x) = ax^2 + bx + c$ in the range. We can make a picture that represents this. Set Cartesian coordinates — the x and y coordinates that people think of as the default — on a surface. Then highlight all the points with coordinates (x, y) which make true the equation $y = f(x)$. This traces out a particular shape, the parabola.

Draw a line that crosses this parabola twice. There’s now one fully-enclosed piece of the surface. How much area is enclosed there? It’s possible to find a triangle with area three-quarters that of the enclosed part. It’s easy to use straightedge and compass to draw a square the same area as a given triangle. Showing the enclosed area is four-thirds the triangle’s area? That can … kind of … be done by straightedge and compass. It takes infinitely many steps to do this. But if you’re willing to allow a process to go on forever? And you show that the process would reach some fixed, knowable answer? This could be done by the ancient Greeks; indeed, it was. Aristotle used this as an example of the method of exhaustion. It’s one of the ideas that reaches toward integral calculus.

This has been a lot of exact, “analytic” results. There are neat numerical results too. Vieta’s formulas, for example, give us good ways to find approximate solutions of the quadratic equation. They work well if one solution is much bigger than the other. Numerical methods for finding solutions tend to work better if you can start from a decent estimate of the answer. And you can learn of numerical stability, and the need for it, studying these.

Numerical calculations have a problem. We have a set number of decimal places with which to work. What happens if we need a calculation that takes more decimal places than we’re given to do perfectly? Here’s a toy version: two-thirds is the number 0.6666. Or 0.6667. Already we’re in trouble. What is three times two-thirds? We’re going to get either 1.9998 or 2.0001 and either way something’s wrong. The wrongness looks small. But any formula you want to use has some numbers that will turn these small errors into big ones. So numerical stability is, in fairness, not something unique to the quadratic equation. It is something you learn if you study the numerics of the equation deeply enough.

I’m also delighted to learn, through Wikipedia, that there’s a prosthaphaeretic method for solving the quadratic equation. Prosthaphaeretic methods use trigonometric functions and identities to rewrite problems. You might call it madness to rely on arctangents and half-angle formulas and such instead of, oh, doing a division or taking a square root. This is because you have calculators. But if you don’t? If you have to do all that work by hand? That’s terrible. But if someone has already prepared a table listing the sines and cosines and tangents of a great variety of angles? They did a great many calculations already. You just need to pick out the one that tells you what you hope to know. I’ll spare you the steps of solving the quadratic equation using trig tables. Wikipedia describes it fine enough.

So you see how much mathematics this connects to. It’s a bit of question-begging to call it that important. As I said, we’ve known the quadratic equation for a long time. We’ve thought about it for a long while. It would be surprising if we didn’t find many and deep links to other things. Even if it didn’t have links, we would try to understand new mathematical tools in terms of how they affect familiar old problems like this. But these are some of the things which we’ve found, and which run through much of what we understand mathematics to be.

The letter ‘R’ for this Fall 2018 Mathematics A-To-Z post should be published Friday. It’ll be available at this link, as are the rest of these glossary posts.

## Silver-Leafed Numbers

In a comment on my “Gilded Ratios” essay fluffy wondered about a variation on the Golden and Golden-like ratios. What’s interesting about the Golden Ratio and similar numbers is that their reciprocal — one divided by them — is a whole number less than the original number. That is, 1 divided by 1.618(etc) is 0.618(etc), which is 1 less than the original number. 1 divided by 2.414(etc) is 0.414(etc), exactly 2 less than the original 2.414(etc). 1 divided by 3.302(etc) is 0.302(etc), exactly 3 less than the original 3.302(etc).

fluffy wondered about a variation. Is there some number x that’s exactly 2 less than 2 divided by x? Or a (presumably) differently number that’s exactly 3 less than 3 divided by it? Yes, there is.

Let me call the whole number difference — the 1 or 2 or 3 or so on, referred to above — by the name b. And let me call the other number — the one that’s b less than b divided by it — by the name x. Then a number x, for which b divided by x is exactly b less than itself, makes true the equation $\frac{b}{x} = x - b$. This is slightly different from the equation used last time, but not very different. Multiply both sides by x, which we know not to be zero, and we get a polynomial.

Yes, quadratic formula, I see you waving your hand in the back there. And you’re right. There are two x’s that will make that equation true. The positive one is $x = \frac12\left( b + \sqrt{b^2 + 4b} \right)$. The negative one you get by changing the + sign, just before the square root, to a – sign, but who cares about that root? Here’s the first several of the (positive) silver-leaf ratios:

Some More Numbers With Cute Reciprocals
Number Silver-Leaf
1 1.618033989
2 2.732050808
3 3.791287847
4 4.828427125
5 5.854101966
6 6.872983346
7 7.887482194
8 8.898979486
9 9.908326913
10 10.916079783
11 11.922616289
12 12.928203230
13 13.933034374
14 14.937253933
15 15.940971508
16 16.944271910
17 17.947221814
18 18.949874371
19 19.952272480
20 20.954451150

Looking over those hypnotic rows of digits past the decimal inspires thoughts. The part beyond the decimal keeps rising, closer and closer to 1. Does it ever get past 1? That is, might (say) the silver-leaf number that’s 2,038 more than its reciprocal be 2,039.11111 (or something)?

No, it never does. There are a couple of ways to prove that, if you feel like. We can take the approach that’s easiest (to my eyes) to imagine. It takes a little algebraic grinding to complete. That is to look for the smallest number b for which the silver-leaf number, $\frac12\left(b + \sqrt{b^2 + 4b}\right)$, is larger than $b + 1$. Follow that out and you realize that it’s any value of b for which 0 is greater than 4. Logically, therefore, we need to take b into a private room and have a serious talk about its job performance, what with it not existing.

A harder proof to imagine working out, but that takes no symbol manipulation, comes from thinking about these reciprocals. Let’s imagine we had some b for which its corresponding silver-leaf number x is more than b + 1. Then, x – b has to be greater than 1. But if x is greater than 1, then its reciprocal has to be less than 1. We again have to talk with b about how its nonexistence is keeping it from doing its job.

Are there other proofs? Most likely. I was satisfied by this point, and resolved not to work on it more until the shower. Updates after breakfast, I suppose.

## Looking to Euler

I haven’t forgotten about writing original material here — actually I’ve been trying to think of why something I’ve not thought about a long while is true, which is embarrassing and hard to do — but in the meanwhile I’d like to remember Leonhard Euler’s 306th birthday and point to Richard Elwes’s essay here about Euler’s totient function. “Totient” is, as best I can determine, a word that exists only for this mathematical concept — it’s the count of how many numbers are relatively prime to a given number — but even if the word comes only from the mildly esoteric world of prime number studies, it’s still one of my favorite mathematical terms. It feels like a word that ought to be more successful. Someday I’ll probably get in a nasty argument with other people playing Boggle about it.

Apparently, though, Euler didn’t dub this quantity the “totient”, and the word is a neologism coined by James Joseph Sylvester (1814 – 1897). That’s pretty respectable company, though: Sylvester — whose name you probably brush up against if you study mathematical matrices — is widely praised for his skill in naming things, although the only terms I know offhand that he gave us were “totient” and “discriminant”. That $b^2 - 4ac$ term in the quadratic formula which tells you whether a quadratic equation has two real, one real, or two imaginary solutions, was a name (not a concept) given by him, and he named (and extended) the similar concept for cubic equations. I do believe there are more such Sylvester-dubbed terms, just, that we need a Wikipedia category to gather them together.

I’m amused to be reminded that, according to the St Andrews biographies of mathematicians, Sylvester at least one tossed off this version of the Chicken McNuggets problem, possibly after he’d worked out the general solution:

I have a large number of stamps to the value of 5d and 17d only. What is the largest denomination which I cannot make up with a combination of these two different values.

## Everything I Learned In Eighth-Grade Math

My title is an exaggeration. In eighth grade Prealgebra I learned many things, but I confess that I didn’t learn well from that particular teacher that particular year. What I most clearly remember learning I picked up from a substitute who filled in a few weeks. It’s a method for factoring quadratic expressions into binomial expressions, and I must admit, it’s not very good. It’s cumbersome and totally useless once one knows the quadratic equation. But it’s fun to do, and I liked it a lot, and I’ve never seen it described as a way to factor quadratic expressions. So let me put it on the web and do what I can to preserve its legacy, and get hundreds of people telling me what it actually is and how everybody but the people I know went through a phase of using it.

It’s a method which looks at first like it’s going to be a magic square, but it’s not, and I’m at a loss what to call it. I don’t remember the substitute teacher’s name, so I can’t use that. I do remember the regular teacher’s name, but it wasn’t, as far as I know, part of his lesson plan, and it’d not be fair to him to let his legacy be defined by one student who just didn’t get him.

## Quadratic Stuff In North Carolina

However weird the linear interpolation of Charlotte, North Carolina’s population may be outside the range from 1970 to 1980, it seems to do nicely enough between those years. And that’s as we might expect, since we used the actual population data from the census days of 1970 and 1980 to form this interpolation. But we don’t have to make a linear interpolation. We could in principle use any function, but let’s try a simple one. This would be a quadratic polynomial, one where the variable x gets raised all the way to the second power, and one that brings back faint memories of the quadratic formula, which is one of the rare pieces of mathematics for which I have a work-related anecdote. Ask sometime if you’re interested.