April 19, 2006
Young Gauss' Trick
Brain Hayes explores the meaning of one of most famous stories in mathematics.
Let me tell you a story, although it's such a well-worn nugget of mathematical lore that you've probably heard it already:(Hat tip: Dan Balis)In the 1780s a provincial German schoolmaster gave his class the tedious assignment of summing the first 100 integers. The teacher's aim was to keep the kids quiet for half an hour, but one young pupil almost immediately produced an answer: 1 + 2 + 3 + ... + 98 + 99 + 100 = 5,050. The smart aleck was Carl Friedrich Gauss, who would go on to join the short list of candidates for greatest mathematician ever. Gauss was not a calculating prodigy who added up all those numbers in his head. He had a deeper insight: If you "fold" the series of numbers in the middle and add them in pairs—1 + 100, 2 + 99, 3 + 98, and so on—all the pairs sum to 101. There are 50 such pairs, and so the grand total is simply 50×101. The more general formula, for a list of consecutive numbers from 1 through n, is n(n + 1)/2.The paragraph above is my own rendition of this anecdote, written a few months ago for another project. I say it's my own, and yet I make no claim of originality. The same tale has been told in much the same way by hundreds of others before me. I've been hearing about Gauss's schoolboy triumph since I was a schoolboy myself.The story was familiar, but until I wrote it out in my own words, I had never thought carefully about the events in that long-ago classroom. Now doubts and questions began to nag at me. For example: How did the teacher verify that Gauss's answer was correct? If the schoolmaster already knew the formula for summing an arithmetic series, that would somewhat diminish the drama of the moment. If the teacher didn't know, wouldn't he be spending his interlude of peace and quiet doing the same mindless exercise as his pupils?
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The excellent two book series "Biographical Encyclopedia of Scientists" by John Daintith, Sarah Mitchell and Elizabeth Tootill, Facts on File, N.Y., 1981 provides excellent brief biographies of the two great contemporary mathematicians Karl Friedrich Gauss ",(b. Apr. 30, 1777 Brunswick, now in West Germany, d., Feb. 23, 1855; Gottingen, now in West Germany)" and Baron Augustin Louis Cauchy, "(b. Aug. 21, 1789; Paris; d. May 23, 1857; Sceaux, France)". In fact, Gauss is credited with "discovering" Cauchy's theorem earlier. In addition to numerous contributions to both mathematics and physics, "Gauss received his doctorate in 1799 from the University of Helmstedt for a proof of the fundamental theorem of algebra, i.e., the theorem that every equation of degree n with complex coefficients has at least one root that is a complex number..."
Reguarding the question posed in the post, my guess is that the teacher was aware of the general formula for the sum. There are now many ways to prove this. Another is to write the sum in reverse; n + (n-1)+ (n-2)...+ 2 +... + (n-n+1).
Because the sum converges since it is finite, the terms may be rearranged so that one may identify nsquared -(S-n) for the sum S. Solving for S yields the well known sum S= n(n+1)/2.
Finite or infinite sums like these can also be expressed as integral representations in terms of a complex variable z, where the sum results from Cauchy's Theorem yielding the series as the sum of the residues at the poles in the z-plane. For example, a common form is sin(pi)z in the denominator which yields poles at the integers along the real axis. One can take different contours around various sections of these poles to obtain various series, finite or infinite. Infinite series are much more complicated due to convergence considerations at infinity. The advantage of this is that a series, finite or infinite with various parameters, can be represented by an integral form, and although some of the parameters may be valid only for certain regions, the integral form may be valid for a greater region in the domain of those parameters, even in their complex planes. This is called analytic continuation. It has been used extensively in elementary particle physics and in the theory or radio wave propagation, or other problems, involving expansions in terms of spherical harmonics. The integral representation is often referred to the "Watson-Sommerfeld Transformation", or, for high energy physics, the Regge Pole representation, after Tullio Regge, the distinguished Italian Mathematician and Physicist who wrote the first papers in Il Nuovo Cimento on the subject in about 1958. This theory has been used to attempt to satisfy the so-called "crossing relations" to calculate the masses of "elementary" particles from first principles. This was before some physicists decided they would be happier with a number of given "elementary" particles called quarks whose masses were assumed a priori, phenomenologically as "inputs". Some of the earlier physicists sought to calculate these "particles" from first principles with as few free parameters as possible. The earlier goal was to avoid as many ad hoc "elementary" objects as possible.
Posted by: Winfield J. Abbe | Apr 19, 2006 8:05:58 PM
As a post script to the above comment, most people, with knowledge of these two remarkable individuals, Gauss and Cauchy, could likely agree on one thing: If such a thing as a genius exists, by virtually any definition of genius, both of these great mathematicians and physicists satisfied the definition.
Yet, notwithstanding this, and their enormous knowledge of algebra, roots of equations, complex analysis, number theory, and mathematical analysis,one basically an "aristocrat", the other sort of from the "other side of the tracks" so to speak, neither could solve Fermat's Last Theorem; namely, x(n)+y(n) = z(n) where x(n) means x raised to the nth power, is impossible in integers where n is a positive integer greater than 2. With all their knowledge about roots, and complex analysis, surely they cast this equation in the form of an nth order polynomial in powers of z say, with x and y regarded as parameters. It is only left then to prove that no solutions of the equation exist if z is a positive integer; or to put it another way, the only solutions in z are complex or non integer solutions. And as we all know, after the arrival of the computer age, no counter example was ever found upon thousands of examples tried. The existing "proof" of Fermat's theorem, produced finally late in the nineteenth century, however, was reached following much more esoteric routes than those likely followed and explored by Gauss and Cauchy much earlier. And if by "proof" one means widely "understood", it is not clear the theorem has yet been "proved" to the intellectual satisfaction either Gauss or Cauchy would have demanded in their day and time.
Posted by: Winfield J. Abbe | Apr 19, 2006 8:48:31 PM
Line -4 above is incorrect and should read "...produced finally late in the twentieth century...". Fermat originally made the claim in seventeenth century, "...that he had a proof of the statement, but no room for it in the margin of his notebook..." or words to that effect. Some people argue he likely did not have a proof. I believe he did, but no one has discovered his proof yet.
A book which summarizes the history of this theorem is "Fermat's Enigma" by Simon Singh, Walker and Co., N.Y., 1997. The book covers the history up to the "proof" of Fermat's Last Theorem by Andrew Wiles in Annals of Mathematics about 1995. But the book is short on details of the proof, or at least details which are comprehensible to any but a small cadre of specialists in the world.
Posted by: Winfield J. Abbe | Apr 19, 2006 9:39:43 PM
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