Calculus is Deeply Irrational

You’re lying on a beautiful beach when you feel a tap on your shoulder, and suddenly you’re not at the beach at all — you’re in a classroom. The student who woke you looks apologetic, and from the front of the room the teacher is staring at you expectantly. He points at the blackboard on which he has written the function f(x) = 6x − x³ next to its graph. “I said: how can we find the maximum value achieved by this function on the interval from 0 to 2?”

A pleasant dream has been replaced by your worst nightmare. But into your still-sleep-fogged conscious mind rises a catchphrase, your only chance for salvation. “Um… Take the derivative and set it equal to zero?”

The professor beams and tells the class “You see? It’s so easy, you can even do it in your sleep!”

This event took place nearly forty years ago (my friend Dan Ullman was a teaching assistant and saw the whole thing, though the function being maximized and the student’s dream were probably different). The story highlights one feature of doing “cookbook-calculus”: you can survive just following the recipes.

But what I want to talk about is what happens next in that classroom, and its broader significance. Because when you differentiate the function 6x − x³ you get the function 6 − 3x², and when you set this expression equal to 0 and simplify you get x² = 2 — an equation that has no solution in the rational numbers.

And the square root of two is nice compared to some of the other beasts that await you in calculus class. If you take the derivative of 10^x, you get 10^x times the irrational number ln 10. Or if you take the derivative of the sine of x (where the angle x is expressed in degrees), you get the sine of x times the irrational number pi/180. The first multiplier, ln 10, whispers “You could make me disappear if only you’d use natural exponentials!” The second multiplier, pi/180, whispers “And you could make me disappear if only you’d express angles in radians!”

But they’re trying to trick you. Accepting natural exponentials means accepting the irrational number e as the One True Base, while accepting radians means worshipping at the altar of the Arch-Irrational, pi. (I jest, of course.)

Why does calculus involve so many irrational numbers? A course in theoretical calculus (also called real analysis) can shed light on the issue. One digs underneath the function-concept to re-explore the number-concept and one finds that the properties of the number-line discussed in high school simply don’t suffice to get the calculus-motor up and running. One needs an extra ingredient in the fuel, called the completeness property of the reals. To paraphrase Obi-Wan Kenobe: The completeness property of the reals is what gives calculus its power. It surrounds the set of real numbers and penetrates it. It binds the number line together.

The real number system, with its plethora of nasty irrational numbers, possesses the completeness property; the tidier rational number system lacks it.

Using the completeness property we can prove the Intermediate Value Theorem: If a continuous function is positive somewhere and negative somewhere, then it’s got to be zero somewhere. Invoking this theorem, we can prove that sqrt(2) exists, since the continuous function x² − 2 is positive for some values of x and negative for others.

Likewise, the completeness property lets us prove the Extreme Value Theorem: A continuous function on a (closed, bounded) interval must achieve its maximum value somewhere. So there must be a place at which the continuous function 6x − x³ achieves its maximum.

If we tried to do calculus using just rational numbers, neither the Intermediate Value Theorem nor the Extreme Value Theorem would be true: for, if we restrict x to rational values, x² − 2 is sometimes positive and sometimes negative but never zero, while 6x − x³ approaches but never achieves a maximum value. (See this Stack Exchange discussion to learn more about what goes wrong with calculus when you try to use only rational numbers.)

There are other theorems in theoretical calculus that work for the real number system but fail for the rational number system: the Mean Value Theorem, the Bolzano-Weierstrass Theorem, the Heine-Borel Theorem, etc.

And now I can get to the real point of this essay. The theorems of the calculus that work for the reals but fail for the rationals, despite the very different claims they make, are all equivalent to each other! That is, if you’re dissatisfied with the rational number system and are looking to upgrade, and you go to the number-system store and say you want a number system that has this feature or that feature, most of the time the sales clerk will reply “Sounds like you want to buy the real numbers.” It’s a package deal; for the price of one feature, you get them all. (See this article to learn more about calculus theorems that are equivalent to the completeness property.)

As I mentioned last month, my favorite theorem-secretly-equivalent-to-completeness is the Constant Value Theorem: it says that if the derivative of a function is 0 everywhere, then the function must be constant. Obvious, right? But if you try do calculus using the rational numbers in place of the real numbers as your “everywhere”, the theorem fails. To see why, look at the function f(x) = sign(x² − 2): it’s +1 where x² > 2 and it’s −1 where x² < 2 (no fair asking “What about where x² = 2?” since we’re restricting to rationals).

In ordinary calculus the function f(x) is discontinuous at x = sqrt(2), but if we’re restricting x to be rational, the function is continuous throughout its domain. In fact, the function is differentiable with derivative 0 throughout its domain. So the truth of the Constant Value Theorem depends on (and in fact is equivalent to) a subtle property of the real numbers that we don’t teach in most calculus classes but which makes calculus work.

Before I end I should probably subvert the jokey title of this essay a bit. The incursion of irrational numbers into mathematics long predates calculus. The ancient Greeks noticed that if you try to measure both a side and a diagonal of a square using the same measuring-stick, you’ll run into trouble. If you use half-a-side as your measuring unit, the diagonal isn’t quite three measuring units long, though the approximation isn’t bad; that’s because the diagonal is slightly less than 3/2 as long as the side. Likewise, if you use a-tenth-of-a-side as your unit, you’ll find that the diagonal is just a hair over fourteen units long; that’s because the diagonal is slightly more than 14/10 as long as the side. There’s no length-unit that does the job exactly, because there’s no fraction whose square is exactly two.

But the Greek mathematician Eudoxus of Cnidus realized that even though the question “For what whole numbers m and n is the diagonal of a square m/n times as long as the side of the square?” has no right answer, each wrong answer is wrong in a consistent way, regardless of which square you’re looking at. That is: whether the square you’re looking at is big or small, the ratio 3-to-2 is always too big and the ratio 14-to-10 is always too small; and so on. Eudoxus realized that being able to compare the square root of two with ordinary numbers like 3/2 and 14/10, and decide in each case which approximations are too big and which are too small, allows us to pin it down with whatever precision we want, and that this ability to compare is good enough not just for all practical purposes, but for all theoretical purposes as well.

Eudoxus’ solution to the problem of irrational numbers (or, expressed in less anachronistic terms, the problem of incommensurable magnitudes) is so satisfactory that most mathematicians dislike the way the mathematical sense of the word “irrational” chafes against its everyday meaning (at least in the languages I’m familiar with). Given that the word “irrational” means “unreasonable”, it’s reasonable for nonmathematicians to infer that irrational numbers must be resistant to human reason, but that natural surmise is a couple of millennia out of date.  A number like the square root of two is only irrational in the sense that it can’t be expressed as a ratio of whole numbers (ir-ratio-nal; get it?). Likewise, calculus isn’t beyond the power of the human mind, though we humans are still working on the problem of how to teach it well to junior members of the species.

But calculus does have many surprises, not least of which is how certain irrational numbers like pi keep popping up as the answer to many seemingly unrelated questions. Perhaps the most surprising thing about calculus, and about mathematics more broadly, is what physicist Eugene Wigner called the “unreasonable effectiveness” of mathematics in describing our universe. To plumb that mystery, we’ll need to find a vista from which we can have a clearer picture of what a different, less reasonable kind of universe might look like. Reason alone won’t take us there.

My Favorite Theorem

2019 is a great year to learn some calculus. Not only are there the videos of Robert Ghrist and Grant Sanderson, but there’s a wonderful new book out by Steven Strogatz. Strogatz has spent the last thirty years growing into the kind of writer who could produce the book about calculus that the world needs, and now he’s produced it. In a few months Ben Orlin will be coming out with a book of his own, and the chapters I’ve seen make me wish I had the power to magically forget calculus (temporarily), so I could have the experience of encountering the subject for the first time through Orlin’s delightful combination of lively prose and cutely inept drawings. And as if that weren’t enough, this year we also have David Bressoud‘s clarion call for teachers to improve the pedagogy of calculus by putting its standard topics back into something like the order in which they were discovered. Calculus is having a gala year.

The celebration is long overdue.¹ Calculus is one of the triumphs of the human spirit, and a demonstration of what perfect straight things (and perfect curvy things) can be made from the crooked timber of humanity. It’s given us a way of seeing order amidst the variety and confusion of reality, hand-in-hand with the expectation that when things happen, they happen for a reason, and that when surprising things happen, it’s time to look for new forces or additional variables.

One of my favorite theorems is a calculus theorem, but it’s not a theorem anyone talks about very much. It may seem mundane (if you’re mathematically sophisticated) or silly (if you’re not). It’s seldom stated, and when it is stated, it’s a lowly lemma, a plank we walk across on the way to our true destination. But it’s a crucial property that holds the real number line together and makes calculus predictive of what happens in the world (as long as we stay away from chaotic and/or quantum mechanical systems). It’s called the Constant Value Theorem, and it can be stated as a succinct motto: “Whatever doesn’t change is constant.” (This is not to be confused with the motto “Change is the only constant”, which happens to be the title of Orlin’s book.) I’ll tell you four things about this theorem that I find surprising and beautiful.

Mathematical Flimflam

[Audio version available at http://mathenchant.org/050.mp3.]

I’m a pure mathematician with no background in applied mathematics. But lately I’ve been striving to make a name for myself in the less-crowded field of mis-applied mathematics, and bogus science more broadly.

Now you may be asking yourself, is bogus science really less crowded a field than good science? After all, if Sturgeon’s law (“Ninety percent of everything is crap”) applies to science, then we can expect crappy science to predominate over the good kind. But bogosity transcends mere crappiness. For something to be bogus, I think there must be an attempt to deceive. Or at least, there must be the appearance of an attempt to deceive. Sometimes the appearance is itself a sham, and that’s the kind of second-order bogosity I enjoy practicing, when I try my hardest to act like someone who genuinely believes (and wants others to believe) a nonsensical theory.

My forum is the Festival of Bad Ad Hoc Hypotheses (BAHFest), held periodically in various locations around the world (San Francisco, Seattle, Cambridge, Sydney, and London). It’s a celebration of well-argued and thoroughly researched but completely incorrect scientific theories. BAHFest is dedicated to the proposition that no matter how absurd a premise is, you can find a way to abuse the tools of science to support your cause and make people laugh in the process. (Or make nerds laugh, anyway.)

BAHFest was the brainchild of Zach Weinersmith whose Infantapulting Hypothesis got the game going.

Carnival of Mathematics #170

I’m hosting issue number 170 because I have a thing for the number’s largest prime factor, but it turns out there’s a reason for a Martin Gardner fan like me to appreciate the number itself: 170 is the number of steps¹ needed to solve a classic mechanical puzzle called The Brain invented by computer scientist Marvin H. Allison, Jr., described by Martin Gardner in his Scientific American essay “The Binary Gray Code”, and still available from Amazon.

The Brain, aka The Brain Puzzle, aka The Brain Puzzler.

Here’s what Gardner says about The Brain:

It consists of a tower of eight transparent plastic disks that rotate horizontally around their centers. The disks are slotted, with eight upright rods going through the slots. The rods can be moved to two positions, in or out, and the task is to rotate the disks to positions that permit all the rods to be moved out. The Gray code supplies a solution in 170 moves.

A Mathematician in the Jury Box, or, “But how should we define ‘intoxicated’?”

Back in the 1990s, when I was serving on a jury in a one-day trial, my mathematical temperament got me in hot water with my fellow jurors; fortunately, my outside-the-classroom mathematical training got me out of it. But that doesn’t come in until the end of the story.

The case featured a couple of surprising twists — which is in itself surprising, since even a single twist is unusual in a one-day trial. It had seemed at first like a very straightforward drunk-driving charge. The defendant went to a party, drank some alcohol, left the party feeling unwell, got into his car, drove off, and blacked out, though with enough advance warning of his impending unconsciousness that he was able to pull over to the side of the road and turn on his hazard lights before passing out. A police officer found him slumped over the wheel of his car. The officer smelled his breath and it smelled of alcohol. The District Attorney presented these facts confidently, as if this was going to be an open-and-shut case. But then, in the kind of surprise you see only on television, the defense attorney asserted (with medical records to support his assertion) that in fact the defendant was diabetic, that someone with diabetes can go into hypoglycemic shock if they ingest a little bit of alcohol on an empty stomach, that the breath of someone in hypoglycemic shock is often nearly indistinguishable from the breath of someone who is drunk, and that the amount of alcohol that the defendant had drunk at the party was, according to witnesses, well under the amount that would cause blood alcohol concentration to reach .08% (the legal definition of “too much”).

Mazes, Puzzles, and Proofs

Many family restaurants offer paper placemats that entice children into solving puzzles as an alternative to kicking each other under the table, blowing bubbles in their beverages, and so on. I remember those placemats from my own childhood and I recall mazes in particular. The mazes weren’t large, but the designers, in their quest to keep us kids occupied as long as possible, would put long dead-ends near the start of the maze. I quickly hit on a strategy that the emphatically named Thomas T. Thomas also found, as he later recounted in his charming essay “Working Backward“: 

One of the most valuable techniques in problem solving I learned in the third grade. But it certainly wasn’t a lesson my teacher intended.

Flip Your Students, Flip Yourself

Collect content-summaries; return them at exams.

I’ll explain that prescription shortly. But first, a confession:

The title of this essay, in which the word “flip” refers to the pedagogical innovation called the “flipped classroom”, is misleading because I’m not going to tell you how to run a flipped classroom. For those of you who haven’t heard, the flipped classroom approach to education is based on moving lecture-delivery out of the classroom and into the dorm room. No, not by having the professor show up in everyone’s dorm room to deliver content on an individual basis; that would be both impractical and improper. Instead, the professor prepares a video, and the students watch it on their laptops. That way, when the students show up in the classroom, they can focus on teacher-supervised activities that take their knowledge to the next level.