A History of Pi
Page 15
This is the result Buffon derived. He also attempted an experimental verification of his result by throwing a needle many times onto ruled paper and observing the fraction of intersections out of all throws. Whether he modified his result for an evaluation of π I do not know, but the problem and its solution were largely forgotten for the next 35 years, until one of the great mathematicians with whom France has been blessed, called attention to it and gave it a new twist.
Pierre Simon Laplace (1749-1827) was one of the “three great L’s” among French mathematicians of the time. The other two, Joseph Louis Lagrange (1736-1813) and Adrien Marie Legendre (1752-1833), were his contemporaries, and all three survived the French Revolution as members of the Committee of Weights and Measures, which discarded the cubits, feet, pounds and miles of the old regime and worked out the metric system as we use it today. It was, incidentally, another mathematician, Lazare Carnot (1753-1823), who saved the young French republic in its hour of greatest need. Scared out of their wits by the cry for liberty, equality and fraternity, Europe’s kings, princes, princelings, counts and whatnots turned on the Revolution. Threatened by internal confusion and the invading armies deep inside France, the Revolution seemed about to be crushed; but Carnot, member of the Committee for Public Safety in charge of military affairs, took command and sent the invaders packing on all fronts, becoming organisateur de la victoire, the hero of the French Revolution. But like so many other sincere revolutionaries after him, Carnot soon observed that a revolution only replaces one tyranny by another, and refusing to go along with its excesses, he was driven into exile as a “royalist.” Significantly, his chair of geometry at the Institut National was unanimously voted to a general; a general by the name of Napoleon Bonaparte, another one in a long line of power-hungry careerists who was to preach liberty and practice oppression.
JOSEPH LOUIS LAGRANGE (1736-1813)
* * *
PIERRE Simon Laplace is known, above all, for authoring two masterpieces, Mecanique céleste (5 vols., 1799-1825) and Théorie analytique des probabilites (1812). The former was the greatest work on celestial mechanics since Newton’s Principia, including many new mathematical techniques, such as the theory of potential. Asked by Napoleon why in the entire work on celestial mechanics he had not once mentioned God, Laplace replied, Sire, je n’avais pas besoin de cette hypothese — Sire, I had no need of that hypothesis. Napoleon, incidentally, appointed Laplace Minister of Interior, but after six weeks dismissed him again, commenting that he “carried the spirit of the infinitely small into the management of affairs.” The Théorie analytique is the foundation of modern probability theory. Among many new mathematical techniques it contains the integral transform that is today the daily bread of every systems engineer and analyst of electrical circuits.
PIERRE SIMON LAPLACE (1749-1827)
It also contains a discussion of Buffon’s problem, which Laplace saw in a new light. From the first and last expressions in (4) we have
and this is an entirely new method of evaluating π: The length of the needle L and the spacing between the lines d are known (usually one makes L = d), and the probability of intersection P can be measured by throwing a needle onto ruled paper a very large number of times, recording the fraction of throws resulting in an intersection of the needle with a line.
This method, which Laplace generalized for paper with two sets of mutually perpendicular lines, has been used by several people as a playful diversion to calculate the first decimal places of π by thousands of throws. One of them was a certain Captain Fox, who indulged in this sport while recovering from wounds incurred in the American Civil War.78
It is not difficult to calculate the probability of obtaining π correct to k decimal places in N throws.79 The results of such a calculation show that this method is very inefficient as far as the numerical computation of π is concerned; for example, the probability of obtaining π correct to 5 decimal places in 3,400 throws of the needle is less than 1.5%, which is very poor.
Nevertheless, Laplace had discovered a powerful method of computation that did not come into its own until the advent of the electronic computer. The method that Laplace proposed consists in finding a numerical value by realizing a random event many times and observing its outcome experimentally. This is today known as a Monte Carlo method (Monte Carlo is the European Las Vegas), and it is used in a wide field of applications ranging from economics to nuclear physics.
Let us first take the example of calculating π by this method. A computer can easily throw a needle 500 times a second, or 1.8 million times per hour. Not literally, of course, but it can be programmed to select a random number (x) for the position of the needle and another (ϕ) for its orientation, which is just as good; it simulates the throwing of a needle. It can also be programmed to observe whether the needle has intersected or not, that is, whether the inequality (1) is satisfied or not. Finally, it is programmed to record the number of intersections in the total number of throws, and after computing the resulting value of π by formula (5), to print the value it has found.
A program of this type is shown on the next page. It is written in BASIC, a simple, but powerful computer language.
The program was actually run80 and resulted in the following values of π in the first 12,000 throws:
We cannot in this way, of course, obtain a better value of π than the one we inserted in line 60 of the program on the opposite page (it occurs there to make the orientation of the needle uniformly distributed between 0° and 180°), and the same line results in an error owing to a series of successive roundings off. However, even if these technicalities were corrected, the result would still be poor, as predicted by the calculation of the probability of obtaining k correct decimal places in a series of n throws. In the same processing time (53 seconds) we could have obtained a much better value, for example, by Euler’s method.
But if the method is not very efficient for calculating π, it is very powerful in other applications. Suppose, for example, that we wish to calculate the mean value of a complicated function of a random variable. This is found by an integration involving the probability density function of the random variable. But sometimes the resulting integral is so complicated that it takes a long time to write the program and that it involves a costly amount of processing time. In that case we do not program the computer for the complicated evaluation of the integral, but we make it simulate the random variable and its function and we make it compute the arithmetic mean of, say, one hundred thousand trials. The result is the required mean value.
Or suppose we wish to find a complicated multiple integral. A Monte Carlo method of finding it (instead of writing a cumbersome program) is to let the computer “shoot” n-tuplets of random numbers. These represent a coordinate in (n-dimensional) space and the coordinate either lies in the volume determined by the integral (“hit”) or it does not (“miss”). Then we let the computer shoot at the target, say, half a million times. The number of hits is then proportional to the n-tuple integral.
The man who taught us to program electronic computers in this way was Pierre Simon Laplace. His computer was neither electronic nor digital. It was an analog computer consisting of one needle and one piece of ruled paper.
16
THE TRANSCENDENCE OF π
Frustra laborant quotquot se calculationibus fatigant pro inventione quadraturae circuli.
Futile is the labor of those who fatigue themselves with calculations to square the circle.
MICHAEL STIFEL (1544)
EULER’S mass annihilation of all problems connected with the evaluation of π gave a complete answer to the question of its numerical value. But he had also opened up a new problem: What kind of number is π? Rational or irrational, algebraic or transcendental? That question was to haunt mathematicians for another 107 years after Euler had asked it.
Already the Greeks before Euclid’s time were familiar with the existence of irrational numbers, that is, of numbers that could not
be expressed as ratios of two integers. They expressed the concept in different words, since the only branch of mathematics with which they were thoroughly familiar was geometry; they said (for example) that the diagonal of a square was incommensurable with its side, the ratio of the two being √2 : 1. Even Aristotle, though shockingly ignorant of the science of his time, was dimly aware of the proof that √2 is irrational. The proof was a reductio ad absurdum and ran as follows:
Let √2, if possible, be equal to a fraction p/q, where the integers p and q are in their lowest terms. Then not more than one of the two integers can be even (or we could cancel by 2). Since 2q2 = p2, it follows that p2, and hence p, must be even (p = 2r), so that q must be odd. But from 2q2 = 4r2 we find q2, and hence q, even; therefore q is both even and odd, which is absurd; hence the assumption √2 = p/q is false.
There is, however, no reason why an irrational number should not be a root of an algenraic equation; for example, √2 is a solution of the algebraic equation x2 – 2 = 0. An algebraic equation is an equation
where n is finite and all coefficients aj are rational (or, which amounts to the same thing, integers, because if they are rational, we can always multiply the equation by a common multiple of the denominators).
By Euler’s time, people began to suspect that there were “worse” numbers than irrational ones, namely numbers that were not only irrational, but that could not even be roots of an algebraic equation. Such numbers are called transcendental.
It was not at all obvious that such numbers exist. For example, the equation
is transcendental, because it is not algebraic; if we expand the sine in a power series, we have
and this is not like (1) because it violates the condition that n must be finite. One of its solutions is x = π/6; but who says that a transcendental equation must have a transcendental solution? The equation sinx = 0 is also transcendental, but one of its solutions, x = 0, is obviously an algebraic number.
The existence of transcendental numbers was not proved until 1840, when Joseph Liouville (1809-1882) showed that one could define numbers (in his proof, as limits of continued fractions) which cannot be roots of any algebraic equation.
But granted that transcendental numbers exist, why should they be of interest? The answer is that they have many interesting properties, and more specifically, that the transcendence of π provides an immediate answer to the age-old problem of whether the circle is squarable.
One of the interesting properties of transcendental numbers is that there are “more” of them than there are algebraic numbers. The “more” is in quotation marks because the set of algebraic numbers is infinite, just as the set of transcendental numbers is infinite. However, there are ways to compare one infinite set to another. When we count a number of objects, say 17 trees in a garden, we are really comparing their amount to the set of natural numbers 1, 2, 3,…, 17. By counting the trees, we are establishing a one-to-one correspondence between each tree and each natural number. We stop the process when this correspondence can no longer be established, in this case for numbers larger than 17, with which no trees can be associated. We then know that the amount of trees in the garden is as large as the amount of numbers in the set of the first 17 natural numbers. But this process of counting can be extended to infinity; any set whose members can be brought into a one-to-one correspondence with the members of the set of natural numbers can be counted this way, even if the counting goes on forever. Such a (countable) set is said to be denumerable, even though it may be infinite. The set of even numbers, for example, is evidently denumerable, so that it is “just as big” as the set of all natural numbers, even or odd. One might have expected it to be only “half as big” as the set of natural numbers, but that is an impermissible generalization of what we are used to in finite sets. The founder of set theory, Georg Cantor (1845-1918), showed that the set of rational, and even of algebraic, numbers is denumerable, but that the set of transcendental (and hence also real) numbers is non-denumerable. These infinites of different orders can be described by new numbers, called cardinal numbers, and these transfinite numbers have their own arithmetic, showing that the joys of mathematics are truly endless: They go beyond infinity.
But the reason why the transcendence of π comes into our story is quite different. We have already seen the reasons why the Greeks insisted that the circle be squared only by a finite number of constructions using compasses and straightedge only (here). After Descartes had found the new geometry, there was the possibility of determining the feasibility of such a construction analytically. A circle can be squared if it can be rectified; if its diameter is unity, we must construct a line of length π. By using compasses and ruler only, we can draw only straight lines and circles, that is, curves whose equations are polynomials of not more than second degree. The points obtained by successive constructions are therefore always intersections (or tangent points) of curves of not more than second degree. We are given a circle whose equation is (we assume unit diameter)
and the final result of the construction is to be a distance equal to π. The coordinates of the end point of this distance is obtained by a chain of constructions, each of which amounts to the following: We are given certain points (from the previous construction) whose coordinates are known numbers; these coordinates (or their simple functions) become the coefficients of the equation that is to be solved in the next step, since an intersection involves the solution of two simultaneous equations.
Starting with (4) and the next step in the construction, characterized by a curve of not more than second degree, we find the intersection of the two curves by solving at most a quadratic equation, whose roots are either rational or irrational involving only square roots. These roots, or their simple functions, become the coefficients of the equation to be solved in the next step of the construction. The next equation is therefore quadratic with coefficients that are either rational or square roots. To convert this to an equation with rational coefficients, it is sufficient (and not even necessary) to square the equation twice over, resulting in an equation of not more than 8th degree. If the construction has s steps, the final equation to be solved in order to yield the length π must therefore be an equation with coefficients of degree not higher than 8s, where s is to be finite.81 If the rectification (or squaring) of the circle is possible in a finite number of quadratic steps, then one of the roots of this algebraic equation is π (or √π); but if π is a number that is not the root of any algebraic equation, then the rectification and squaring of the circle (by Greek rules) is impossible.
Thus, the question of whether the circle can be squared by Euclidean geometry could be answered as follows: If π is a transcendental number, then it cannot be done.
The theory of equations thus had, as modern jargon goes, “put a handle on the problem” which Anaxagoras had pondered on the floor of his prison in Athens in the 5th century B.C. When Lindemann in 1882 finally proved that π is transcendental, he finished off the prey with the last blow; but it was the above line of reasoning that first brought it to bay. Although it cannot be attributed to any single man, the mathematician to whom the theory of equations owes more than to anybody else was Carl Friedrich Gauss (1777-1855).
Gauss also estimated the value of π by using lattice theory and considering a lattice inside a large circle,83 but this is as close as this great mathematician came to the story of π.
* * *
LONG before Liouville’s proof of the existence of transcendental numbers (1840), the irrationality of π was established by the Swiss mathematician Johann Heinrich Lambert (1728-1777), and by Adrien-Marie Legendre (1752-1833). Lambert proved the irrationality of π in 1767, but Legendre provided a more rigorous proof of an auxiliary theorem concerning continued fractions that Lambert had used. In his treatise Vorläufige Kenntnisse für die, so die Quadratur und Rectification des Circuls suchen (Preliminary knowledge for those who seek the quadrature and rectification of the circle, 1767), Lambert investigated certain
continued fractions and proved the following theorem:
If x is a rational number other than zero, then tan x cannot be rational.
From this it immediately follows that
If tan x is rational, then x must be irrational or zero (for if it were not so, the original theorem would be contradicted).
Since tan(π/4) = 1 is rational, π/4 must be irrational, and hence the irrationality of π is established.
Lambert also gave an interesting continued fraction for π,
The inverse convergents of this continued fraction are shown on the opposite page. The first convergents of Lambert’s continued fraction had been found in some way or other long before his time:
3
value implied by I Kings vii, 23;
22/7
upper bound found by Archimedes, 3rd century B.C.;
333/106
lower bound found by Adriaan Anthoniszoon, ca. 1583;
355/113
found by Valentinus Otho, 1573;84 also Anthoniszoon, Metius and Viète (all 16th century).
Inverse convergents given by Lambert.
Legendre, in his Elements de Géometrie (1794) proved the irrationality of π more rigorously, and also gave a proof that π2 is irrational, dashing the hopes that π might be the square root of a rational number. Toward the end of his investigation, Legendre writes: “It is probable that the number π is not even contained among the algebraic irrationalities, i.e., that it cannot be the root of an algebraic equation with a finite number of terms, whose coefficients are rational. But it seems to be very difficult to prove this strictly.”