Math Horizons - Volume XXVIII, Issue 3, FEBRUARY 2021
The Square Root of Pi
James Propp 2021-03-01 06:28:22
march 14 (month 3, day 14) is the day math nerds celebrate the number p (3.14…), and you might be one of them. But if you’re getting tired of your p served plain, why not spice things up by combining the world’s favorite nerdy number with the world’s favorite nerdy operation?
The square root of π has attracted attention for almost as long as π itself. When you’re an ancient Greek mathematician studying circles and squares and playing with straightedges and compasses, it’s natural to try to find a circle and a square that have the same area. If you start with the circle and try to find the square, that’s called squaring the circle. A circle of radius r = 1 has area π .12 = π so a square with side length s has the same area as the circle if s2 = p, that is, if s = √π.
It’s well known that squaring the circle is impossible in the sense that, if you use the classic Greek tools in the classic Greek manner, you can’t construct a square whose side-length is √π. But what’s less well known is that there are (at least!) two other places in mathematics where the square root of π crops up: an infinite product that on its surface makes no sense and a calculus problem with a solution involving a surface.
A Nonsensical Factorial
The factorial provides a handy shortcut in many counting problems. If someone asks you how many different ways there are to order the 52 cards of a standard deck, the answer is “52 factorial” (meaning 52 × 51 × 50 × … × 3 × 2 × 1, often written as “52!”); this answer takes a lot less time to say than “eighty unvigintillion, six hundred and fifty-eight vigintillion, …, eight hundred and twenty-four trillion”. But how should we define n! if n is not a counting number? What if, say, n is negative one-half?
A practical-minded person might gibe that someone who wants to know how many ways there are to order a stack consisting of negative one-half cards isn’t playing with a full deck. But this kind of craziness is often surprisingly fruitful; the symbolisms mathematicians come up with sometimes take on a life of their own, and living things want to grow. Or, as the great mathematician Leonhard Euler wrote (unless he didn’t), “My pen is a better mathematician than I am,” meaning that notations sometimes precede understanding.
By trying to extend the factorial function to noncounting numbers like −1/2, Euler invented the gamma function, which is important in both pure and applied mathematics. Specifically, Euler showed that when n is a positive integer, n! is equal to the area bounded by the vertical axis x = 0, the horizontal axis y = 0, and the graph of the curve y = xne−x, which is described by the integral
Here, e = 2.718… is another famous nerdy mathematical constant, dominating the world of exponentials and logarithms the way π rules the world of sines and cosines. The cool thing about the integral expression is that it makes sense when you replace n by −1/2 (corresponding to the area under the curve y = x−1/2e−x in figure 1) and gives the value π.
What’s more, if you find other integrals that coincide with n! when n is a positive integer, they’ll give you p when you replace n by −1/2. Metaphorically, you could say that even though (−1 / 2)×(−3 / 2)×(−5 / 2)×...×3×2×1 isn’t equal to any number, the number it’s trying to equal is √π.
A Nonsensical Sum
Let’s consider a simpler example of an expression that’s “trying to equal” something. If r is a positive number that’s less than 1, then the infinite sum 1−r +r2 −r3 +... (an alternating geometric series) converges to 1 / (1 + r) in the sense that the partial sums 1, 1 − r, 1 – r + r2, 1 – r + r2 − r3, … get ever closer to 1 / (1 + r) as the number of terms gets ever bigger. Now, when r is 1, the expression 1 / (1 + r) is defined and equals 1/2, but the alternating sum 1−1+1−1+… does not get ever closer to 1/2 or to any other number. The partial sums 1, 1−1, 1−1+1, 1−1+1−1, … just vacillate between two values, 1, 0, 1, 0, …, instead of converging.
But what if we change the definition of convergence? Ernesto Cesàro proposed a more permissive definition in which we take a sequence of numbers that annoyingly refuses to converge and replace it by a new, more tractable sequence whose nth term is the average of the first n terms of the original sequence.
Applying Cesàro’s procedure to the sequence 1, 0, 1, 0, …, we get 1/1, (1+0)/2, (1+0+1)/3, (1+0+1+0)/4, …. Sure enough, the new sequence converges to 1/2, exactly what we obtained from 1 / (1 + r) when we replaced r by 1.
So I’ve shown you two different shady methods of assigning a value to the nonconverging sum 1 – 1 + 1 – 1 + …. One method, called Abel summation, considers the more general sum 1 – r + r2 – r3 + …, finds an expression for the sum that’s valid for all positive r < 1, and substitutes r = 1. The other method replaces the ordinary notion of summation taught in calculus classes by Cesàro summation.
And here’s an even simpler shady proof: define x = 1 − 1 + 1 − 1 + …. Then 1 − x = 1 − (1 − 1 + 1 − …) = 1 − 1 + 1 − 1 + … = x. Thus, 1 − x = x, yielding x = 1/2.
What’s interesting is that even though these shady methods are shady in different ways, they all give the same value. This phenomenon— different shady methods arriving at the same answer—crops up a lot at the frontiers of math, and it often points to mathematical concepts that have not yet come into view. When we finally climb the hill that hid those concepts from view, we find that there’s nothing illegitimate about those concepts; they’re just more mathematically technical than, and not as attention-grabbing as, a formula like “1−1+1−1+… = 1/2”. Such a formula, presented out of context, with all the sense-making scaffolding yanked away, is sort of like Lewis Carroll’s Cheshire cat: all that’s left is a (possibly infuriating) smirk.
Infamous examples of mathematical YouTube sensationalism are the numerous videos showing the formula 1 + 2 + 3 + … = −1/12 (and its less celebrated sibling 1 + 1 + 1 + … = −1/2). When equations like these are hyped outside of their proper context, math fans sometimes respond with anger: “Those bastards are changing the rules again!”
What often isn’t explained (because it’s fairly technical) is what the new rules are. What can and should be explained is that this rules-change is analogous to what you’d see if you were an American watching American football on TV and then changed the channel to watch Australian football. Well-educated sports fans don’t fume, “They’re playing it wrong! Why, even my gym teacher knows more about football than these players do!”; they recognize that the Australians are playing a different sport. Same here, except that the name of the sport is “zeta-function regularization,” which is played with numbers, and the players are analytic number theorists, only some of whom are Australian.
Lord Kelvin’s Definition of a Mathematician
The physicist William Thomson, better known as Lord Kelvin, was a big fan of mathematics, calling it “the etherealization of common sense” and “the only good metaphysics.” According to an anecdote recounted by his biographer S. P. Thompson, Kelvin was a bit of an awestruck fanboy when it came to mathematicians themselves:
Once when lecturing to a class he [Lord Kelvin] used the word “mathematician,” and then interrupting himself asked his class: “Do you know what a mathematician is?” Stepping to the blackboard he wrote upon it:
Then putting his finger on what he had written, he turned to his class and said: “A mathematician is one to whom that is as obvious as that twice two makes four is to you.”
This formula (usually attributed to Gauss) is important and fairly well known, but I don’t think I know a single mathematician who regards the formula as obvious, so I don’t think Kelvin’s definition of a mathematician is a good one. Here’s an alternative definition for your consideration: A mathematician is one who recognizes the difference between what is obvious and what is merely familiar. Or: A mathematician is one who recognizes the difference between what is obvious and what one has come to understand in stages, by means of a nontrivial chain of trivial steps.
This seems like an ungrateful way to treat a scientist who, when all is said and done, was just trying to praise my kind. But a mathematician is one to whom flattery is annoying if it is inaccurate. (See, I can overgeneralize too!)
The expression
represents the area under the curve y = e-x2 as shown in figure 4. The ∫ notation, which we saw earlier, is due to Leibniz, who chose a stylized version of the letter “s” to commemorate the fact that the way we can compute such areas is by first approximating them as sums, and by then atoning for the error of the approximation by using ever better approximations and seeing what the approximations converge to.
Solving a Problem by Making it Simpler
Let’s talk about sums for a minute. Specifically, consider the problem of adding together all the numbers in the first four rows and first four columns of the multiplication table:
A trick for computing the sum is to consider that the product (1 + 2 + 3 + 4) × (1 + 2 + 3 + 4), if expanded out by the general distributive law, would have the 16 numbers, 1×1, 1×2, …, 4×3, and 4×4, as its constituent terms, so that the sum of the sixteen numbers must be 10 × 10, or 100. (“A mathematician is someone who works hard at being lazy,” said Murray Klamkin, riffing on George Pólya.)
You can apply the same idea to sum the numbers in the infinite array
by expanding (1 + 1/2 + 1/4 + 1/8 + …) multiplied by itself. The infinite sum 1 + 1/2 + 1/4 + 1/8 +… equals 2 (i.e., the partial sums converge to 2) so the sum of the numbers in the infinite array equals 2 times 2, and as Lord Kelvin said, “twice two makes four.” Note that we can also group the numbers in the table by diagonals to get 1×1 + 2×1/2 + 3×1/4 + 4×1/8 + …. So the infinite sum 1/1 + 2/2 + 3/4 + 4/8 + 5/16 + … (where successive numerators increase by 1 and successive denominators double) equals 4. A mathematician is someone who thinks this is cute. Not “obvious”—just cute.
This way of summing an array of numbers is the sort of trick you see quite often in mathematics, where you reduce a two-dimensional problem to a one-dimensional problem, or more broadly solve a problem by reducing it to something that looks easier. But how often do you solve a problem by “reducing” it to something that looks harder? That counterintuitive tactic provides the nicest way I know to prove the famous formula of Gauss that Lord Kelvin quoted.
Solving a Problem by Complicating It
Let A be the area between the x-axis and the curve y = e−x2 represented by Gauss’s integral. Then multivariate calculus can be used to show that A2 is the volume between the xy-plane and the surface z = e−−x2−e−y2 pictured in figure 5.
Using the laws of exponents, we can rewrite the surface as z = e−(x +y ). 2 2 This crucial step reveals that the surface has a surprising symmetry: given two points (x,y) and (a,b) that are equal distance, say r, from the origin, we have x2 + y2 = r2 = a2 +b2 so that e−(x +y) = e−(a +b), 2 2 2 2 which tells us that the three-dimensional graph of the function e−(x2+y2) has rotational symmetry around the z-axis. One thing I love about the proof of Gauss’s formula is the way it brings together so much mathematics: e, p, and even the Pythagorean theorem!
The rotational symmetry enables calculus students to treat the space between the xy-plane and the surface z = e−(x2+y2) as a solid of revolution, allowing them to compute the volume using the method of cylindrical shells. One gets A2 = π, from which the formula on Lord Kelvin’s blackboard follows.
I would modify Kelvin’s adage to say that a mathematician is someone who, having learned the preceding derivation, may not be able to remember the details, but remembers that circles play a role, and that the value of the integral therefore involves π. Or perhaps a mathematician is someone who finds the proof beautiful, irrespective of the beauty of the formula itself.
Gauss’ formula isn’t just a mathematical curiosity: the equation y = e−x2 is closely related to the famous bell-shaped curve of statistics, and the fact that the area beneath it is p explains where the p in basic statistical formulas comes from.
To revisit the classic problem with which this essay began: You can’t square the circle, but you can do something much more important, namely, you can prove
by interpreting the square of the left-hand side as a volume and then computing that volume using circles.
To give Lord Kelvin his due, let me credit him for praising mathematicians for conceptual understanding, even if his praise strikes me as excessive. His adage is a lot better than something along the lines of “A mathematician is someone who knows the first half-dozen digits of the square root of π. ”
Euler’s Gamma Function
Euler actually defined the gamma function G(z) as which leads to a slightly annoying off-by-one issue: instead of Γ(n) =n! we have Γ(n) =(n−1)! . Another issue worth mentioning is that Euler’s definition only works for positive values of z. The area under the curve becomes infinite when z ≤ 0. Fortunately, a technique called analytic continuation can be used to figure out what expressions like Γ(−1 / 2) are “trying” to equal.
It’s not too hard to guess what we might want Γ(−1 / 2) to equal: the formula n! = n(n−1)! corresponds to the formula Γ(z +1) = zΓ(z). Applying this equation with z = −1/2 gives
The values of G(z) for Γz = 1/2, 3/2, etc. play a role in the formula for the volume of an n-dimensional ball of radius R, which is given by Rnπn/2 / Γ(n / 2+1). Curiously, plugging in n = −1 tells us that the “volume of a (−1)-dimensional ball” of radius R is trying to equal 1 / πR, but I have no idea what this might mean!
*James Propp (@JimPropp) teaches math at UMass Lowell and does research in combinatorics, probability, and dynamical systems. He blogs at mathenchant.wordpress.com. He would like to thank to Henry Baker, Michael Collins, David Feldman, Sandi Gubin, David Jacobi, Kerry Mitchell, Cris Moore, Ben Orlin, Evan Romer, James Tanton, and Glen Whitney. Read an extended version of this essay at http://bit.ly/3gNC9Ra.
©Taylor and Francis. View All Articles.