and if the hypotenuse were 7, its square would be 7 × 7 = 49.
Or if you make the legs 12 units long, the hypotenuse is almost exactly 17 units, but is tantalizingly too short, because 122 + 122 is 288, a smidgen less than 172, which is 289.
And at some point around the fifth century BCE, a member of the Pythagorean school made a shocking discovery: there was no way to measure the isosceles right triangle so that the length of each side was a whole number. Modern people would say “the square root of 2 is irrational”—that is, it is not the ratio of any two whole numbers. But the Pythagoreans would not have said that. How could they? Their notion of quantity was built on the idea of proportions between whole numbers. To them, the length of that hypotenuse had been revealed to be not a number at all.
This caused a fuss. The Pythagoreans, you have to remember, were extremely weird. Their philosophy was a chunky stew of things we’d now call mathematics, things we’d now call religion, and things we’d now call mental illness. They believed that odd numbers were good and even numbers evil; that a planet identical to our own, the Antichthon, lay on the other side of the sun; and that it was wrong to eat beans, by some accounts because they were the repository of dead people’s souls. Pythagoras himself was said to have had the ability to talk to cattle (he told them not to eat beans) and to have been one of the very few ancient Greeks to wear pants.
The mathematics of the Pythagoreans was inseparably bound up with their ideology. The story (probably not really true, but it gives the right impression of the Pythagorean style) is that the Pythagorean who discovered the irrationality of the square root of 2 was a man named Hippasus, whose reward for proving such a nauseating theorem was to be tossed into the sea by his colleagues, to his death.
But you can’t drown a theorem. The Pythagoreans’ successors, like Euclid and Archimedes, understood that you had to roll up your sleeves and measure things, even if this brought you outside the pleasant walled garden of the whole numbers. No one knew whether the area of a circle could be expressed using whole numbers alone.* But wheels must be built and silos filled;* so the measurement must be done.
The original idea comes from Eudoxus of Cnidus; Euclid included it as book 12 of the elements. But it was Archimedes who really brought the project to its full fruition. Today we call his approach the method of exhaustion. And it starts like this.
The square in the picture is called the inscribed square; each of its corners just touches the circle, but it doesn’t extend beyond the circle’s boundary. Why do this? Because circles are mysterious and intimidating, and squares are easy. If you have before you a square whose side has length X, its area is X times X—indeed, that’s why we call the operation of multiplying a number by itself squaring! A basic rule of mathematical life: if the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object.
The inscribed square breaks up into four triangles, each of which is none other than the isosceles triangle we just drew.* So the square’s area is four times the area of the triangle. That triangle, in turn, is what you get when you take a 1 x 1 square and cut it diagonally in half like a tuna fish sandwich.
The area of the tuna fish sandwich is 1 × 1 = 1, so the area of each triangular half-sandwich is 1/2, and the area of the inscribed square is 4 times 1/2, or 2.
By the way, suppose you don’t know the Pythagorean Theorem. Guess what—you do now! Or at least you know what it has to say about this particular right triangle. Because the right triangle that makes up the lower half of the tuna fish sandwich is exactly the same as the one that is the northwest quarter of the inscribed square. And its hypotenuse is the inscribed square’s side. So when you square the hypotenuse, you get the area of the inscribed square, which is 2. That is, the hypotenuse is that number which, when squared, yields 2; or, in the usual more concise lingo, the square root of 2.
The inscribed square is entirely contained within the circle. If its area is 2, the area of the circle must be at least 2.
Now we draw another square.
This one is called the circumscribed square; it, too, touches the circle at just four points. But this square contains the circle. Its sides have length 2, so its area is 4; and so we know the area of the circle is at most 4.
To have shown that pi is between 2 and 4 is perhaps not so impressive. But Archimedes is just getting started. Take the four corners of your inscribed square and mark new points on the circle halfway between each adjacent pair of corners. Now you’ve got eight equally spaced points, and when you connect those, you get an inscribed octagon, or, in technical language, a “stop sign”:
Computing the area of the inscribed octagon is a bit harder, and I’ll spare you the trigonometry. The important thing is that it’s about straight lines and angles, not curves, and so it was doable with the methods available to Archimedes. And the area is twice the square root of 2, which is about 2.83.
You can play the same game with the circumscribed octagon
whose area is 8(√2 − 1), a little over 3.31.
So the area of the circle is trapped in between 2.83 and 3.31.
Why stop there? You can stick points in between the corners of the octagon (whether inscribed or circumscribed) to make a 16-gon; after some more trigonometric figuring, that tells you that the area of the circle is in between 3.06 and 3.18. Do it again, to make a 32-gon; and again, and again, and pretty soon you have something that looks like this:
Wait, isn’t that just the circle? Of course not! It’s a regular polygon with 65,536 sides. Couldn’t you tell?
The great insight of Eudoxus and Archimedes was that it doesn’t matter whether it’s a circle or a polygon with very many very short sides. The two areas will be close enough for any purpose you might have in mind. The area of the little fringe between the circle and the polygon has been “exhausted” by our relentless iteration. The circle has a curve to it, that’s true. But every tiny little piece of it can be well approximated by a perfectly straight line, just as the tiny little patch of the earth’s surface we stand on is well approximated by a perfectly flat plane.*
The slogan to keep in mind: straight locally, curved globally.
Or think of it like this. You are streaking downward toward the circle as from a great height. At first you can see the whole thing:
Then just one segment of arc:
And a still smaller segment:
Until, zooming in, and zooming in, what you see is pretty much indistinguishable from a line. An ant on the circle, aware only of his own tiny immediate surroundings, would think he was on a straight line, just as a person on the surface of the earth (unless she is clever enough to watch objects crest the horizon as they approach from afar) feels like she’s standing on a plane.
THE PAGE WHERE I TEACH YOU CALCULUS
I will now teach you calculus. Ready? The idea, for which we have Isaac Newton to thank, is that there’s nothing special about a perfect circle. Every smooth curve, when you zoom in enough, looks just like a line. Doesn’t matter how winding or snarled it is—just that it doesn’t have any sharp corners.
When you fire a missile, its path looks like this:
The missile goes up, then down, in a parabolic arc. Gravity makes all motion curve toward the earth; that’s among the fundamental facts of our physical life. But if we zoom in on a very short segment, the curve starts to look like this:
And then like this:
Just like the circle, the missile’s path looks to the naked eye like a straight line, progressing upward at an angle. The deviation from straightness caused by gravity is too small to see—but it’s still there, of course. Zooming in to an even smaller region of the curve makes the curve even more like a straight line. Closer and straighter, closer and straighter . . .
Now here’s the conceptual leap. Newton sai
d, look, let’s go all the way. Reduce your field of view until it’s infinitesimal—so small that it’s smaller than any size you can name, but not zero. You’re studying the missile’s arc, not over a very short time interval, but at a single moment. What was almost a line becomes exactly a line. And the slope of this line is what Newton called the fluxion, and what we’d now call the derivative.
That’s a kind of jump Archimedes wasn’t willing to make. He understood that polygons with shorter sides got closer and closer to the circle—but he would never have said that the circle actually was a polygon with infinitely many infinitely short sides.
Some of Newton’s contemporaries, too, were reluctant to go along for the ride. The most famous objector was George Berkeley, who denounced Newton’s infinitesimals in a tone of high mockery sadly absent from current mathematical literature: “And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”
And yet calculus works. If you swing a rock in a loop around your head and suddenly release it, it’ll shoot off along a linear trajectory at constant speed,* exactly in the direction that calculus says the rock is moving at the precise moment you let go. That’s yet another Newtonian insight; objects in motion tend to proceed in a straight-line path, unless some other force intercedes to nudge the object one way or the other. That’s one reason linear thinking comes so naturally to us: our intuition about time and motion is formed by the phenomena we observe in the world. Even before Newton codified his laws, something in us knew that things like to move in straight lines, unless given a reason to do otherwise.
EVANESCENT INCREMENTS AND UNNECESSARY PERPLEXITIES
Newton’s critics had a point; his construction of the derivative didn’t amount to what we’d call rigorous mathematics nowadays. The problem is the notion of the infinitely small, which was a slightly embarrassing sticking point for mathematicians for thousands of years. The trouble started with Zeno, a fifth-century-BCE Greek philosopher of the Eleatic school who specialized in asking innocent-seeming questions about the physical world that inevitably blossomed into huge philosophical brouhahas.
His most famous paradox goes like this. I decide to walk to the ice cream store. Now certainly I can’t get to the ice cream store until I’ve gone halfway there. And once I’ve gone halfway, I can’t get to the store until I’ve gone half the distance that remains. Having done so, I still have to cover half the remaining distance. And so on, and so on. I may get closer and closer to the ice cream store—but no matter how many steps of this process I undergo, I never actually reach the ice cream store. I am always some tiny but nonzero distance away from my two scoops with jimmies. Thus, Zeno concludes, to walk to the ice cream store is impossible. The argument works just as well for any destination: it’s equally impossible to walk across the street, or to take a single step, or to wave your hand. All motion is ruled out.
Diogenes the Cynic was said to have refuted Zeno’s argument by standing up and walking across the room. Which is a pretty good argument that motion is actually possible; so something must be wrong with Zeno’s argument. But where’s the mistake?
Break down the trip to the store numerically. First you go halfway. Then you go half of the remaining distance, which is 1/4 of the total distance, and you’ve got 1/4 left to go. So half of what’s left is 1/8, then 1/16, then 1/32. Your progress toward the store looks like this:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + . . .
If you add up ten terms of this sequence you get about 0.999. If you add up twenty terms it’s more like 0.999999. In other words, you are getting really, really, really close to the store. But no matter how many terms you add, you never get to 1.
Zeno’s paradox is much like another conundrum: is the repeating decimal 0.99999. . . . . . equal to 1?
I have seen people come nearly to blows over this question.* It’s hotly disputed on websites ranging from World of Warcraft fan pages to Ayn Rand forums. Our natural feeling about Zeno is “of course you eventually get your ice cream.” But in this case, intuition points the other way. Most people, if you press them, say 0.9999 . . . doesn’t equal 1. It doesn’t look like 1, that’s for sure. It looks smaller. But not much smaller! Like Zeno’s hungry ice cream lover, it gets closer and closer to its goal, but never, it seems, quite makes it there.
And yet, math teachers everywhere, myself included, will tell them, “No, it’s 1.”
How do I convince someone to come over to my side? One good trick is to argue as follows. Everyone knows that
0.33333. . . . . = 1/3.
Multiply both sides by 3 and you’ll see
0.99999. . . . = 3/3 = 1.
If that doesn’t sway you, try multiplying 0.99999 . . . by 10, which is just a matter of moving the decimal point one spot to the right.
10 × (0.99999 . . .) = 9.99999. . . .
Now subtract the vexing decimal from both sides:
10 × (0.99999 . . .) − 1 × (0.99999 . . .) = 9.99999 . . . − 0.99999. . . . .
The left-hand side of the equation is just 9 × (0.99999 . . .), because 10 times something minus that something is 9 times the aforementioned thing. And over on the right-hand side, we have managed to cancel out the terrible infinite decimal, and are left with a simple 9. So we end up with
9 × (0.99999 . . .) = 9.
If 9 times something is 9, that something just has to be 1—doesn’t it?
These arguments are often enough to win people over. But let’s be honest: they lack something. They don’t really address the anxious uncertainty induced by the claim 0.99999 . . . = 1; instead, they represent a kind of algebraic intimidation. “You believe that 1/3 is 0.3 repeating—don’t you? Don’t you?”
Or worse: maybe you bought my argument based on multiplication by 10. But how about this one? What is
1 + 2 + 4 + 8 + 16 + . . . ?
Here the “. . .” means “carry on the sum forever, adding twice as much each time.” Surely such a sum must be infinite! But an argument much like the apparently correct one concerning 0.9999 . . . seems to suggest otherwise. Multiply the sum above by 2 and you get
2 × (1 + 2 + 4 + 8 + 16 + . . .) = 2 + 4 + 8 + 16 + . . .
which looks a lot like the original sum; indeed, it is just the original sum (1 + 2 + 4 + 8 + 16 + . . .) with the 1 lopped off the beginning, which means that 2 × (1 + 2 + 4 + 8 + 16 + . . .) is 1 less than (1 + 2 + 4 + 8 + 16 + . . .). In other words,
2 × (1 + 2 + 4 + 8 + 16 + . . .) − 1 × (1 + 2 + 4 + 8 + 16 + . . .) = −1.
But the left-hand side simplifies to the very sum we started with, and we’re left with
1 + 2 + 4 + 8 + 16 + . . . = −1.
Is that what you want to believe?* That adding bigger and bigger numbers, ad infinitum, flops you over into negativeland?
More craziness: What is the value of the infinite sum
1 − 1 + 1 − 1 + 1 − 1 + . . .
One might first observe that the sum is
(1 − 1) + (1 − 1) + (1 − 1) + . . . = 0 + 0 + 0 + . . .
and argue that the sum of a bunch of zeroes, even infinitely many, has to be 0. On the other hand, 1 − 1 + 1 is the same thing as 1 − (1 − 1), because the negative of a negative is a positive; applying this fact again and again, we can rewrite the sum as
1 − (1 − 1) − (1 − 1) − (1 − 1) . . . = 1 − 0 − 0 − 0 . . .
which seems to demand, in the same way, that the sum is equal to 1! So which is it, 0 or 1? Or is it somehow 0 half the time and 1 half the time? It seems to depend where you stop—but infinite sums never stop!
Don’t decide yet, because it gets worse. Suppose T is the value of our mystery sum:
T = 1 − 1 + 1 − 1 + 1 − 1 + . . .
Taking the negativ
e of both sides gives you
−T = −1 + 1 − 1 + 1 . . .
But the sum on the right-hand side is precisely what you get if you take the original sum defining T and lop off that first 1, thus subtracting 1; in other words,
−T = −1 + 1 − 1 + 1 . . . = T − 1.
So −T = T − 1, an equation concerning T which is satisfied only when T is equal to 1/2. Can a sum of infinitely many whole numbers somehow magically become a fraction? If you say no, you have the right to be at least a little suspicious of slick arguments like this one. But note that some people said yes, including the Italian mathematician/priest Guido Grandi, after whom the series 1 − 1 + 1 − 1 + 1 − 1 + . . . is usually named; in a 1703 paper, he argued that the sum of the series is 1/2, and moreover that this miraculous conclusion represented the creation of the universe from nothing. (Don’t worry, I don’t follow that last step either.) Other leading mathematicians of the time, like Leibniz and Euler, were on board with Grandi’s strange computation, if not his interpretation.
But in fact, the answer to the 0.999 . . . riddle (and to Zeno’s paradox, and to Grandi’s series) lies a little deeper. You don’t have to give in to my algebraic strong-arming. You might, for instance, insist that 0.999 . . . is not equal to 1, but rather 1 minus some tiny infinitesimal number. And, for that matter, you might further insist that 0.333 . . . is not exactly equal to 1/3, but also falls short by an infinitesimal quantity. This point of view requires some stamina to push through to completion, but it can be done. I once had a calculus student named Brian who, unhappy with the classroom definitions, worked out a fair chunk of the theory by himself, referring to his infinitesimal quantities as “Brian numbers.”
Brian was not actually the first to get there. There’s a whole field of mathematics that specializes in contemplating numbers of this kind, called nonstandard analysis. The theory, developed by Abraham Robinson in the mid-twentieth century, finally made sense of the “evanescent increments” that Berkeley found so ridiculous. The price you have to pay (or, from another point of view, the reward you get to reap) is a profusion of novel kinds of numbers; not only infinitely small ones, but infinitely large ones, a huge spray of them in all shapes and sizes.*
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 4