If you examine the graphic above, you can see that if you add the first and last terms you get Remember that in an arithmetic series, the common difference is constant and this pattern of adding and subtracting the same value as the terms are paired will continue. All sums will be Carl Friedrich Gauss was a German mathematician who contributed in many fields of mathematics and science and is touted as one of history's most influential mathematicians.
All of Gauss' wrapped pairs have a sum of While any pair could be used, it will be easier if we choose the first pair on the left, the addition of the first and last term, as these terms are usually more readily available.
Gauss multiplied this sum times 50, which is HALF the number of terms in his sequence. So, we need to multiply times half of the number of terms in the sequence, which is represented by n. In the literature I have surveyed, the series makes its first appearance in , some 80 years after Sartorius wrote his memoir. The example is introduced in a biography of Gauss by Ludwig Bieberbach a mathematician notorious as the principal instrument of Nazi anti-Semitism in the German mathematical community.
Bieberbach's telling of the story is also the earliest I have seen to specify Gauss's strategy for calculating the sum—the method of forming pairs that add to Should Bieberbach therefore be regarded as the source from whom scores of later authors have borrowed these "facts"? Or is this a case of multiple independent invention? If you think it utterly implausible that two or more authors would come up with the same example and the same method, then Bieberbach himself is disqualified as the source.
A catalog of stories records features of some 70 tellings of the Gauss anecdote. Some of these features, such as the busy-work theme, were not present in the original versions but are now commonplace. A more-complete version of this table, including links to the original sources, is available here. Furthermore, in the years since Bieberbach wrote, there is unmistakable evidence of independent invention.
Not all versions agree that the sequence of numbers was the set of consecutive integers from 1 through Although that series is the overwhelming favorite, many others have been proposed. Some are slight variations: or Several authors seem to feel that adding up numbers is too big a job for primary-school students, and so they trim the scope of the assignment, suggesting , or , or , or , or A few others apparently think that is too easy, and so they give , or else a series in which the difference between successive terms is a constant other than 1, such as the sequence 3, 7, 11, 15, 19, 23, The example series chosen by various authors and other features of the versions are tabulated in the table above.
Perhaps the most influential version of the story after that of Sartorius is the one told by Eric Temple Bell in Men of Mathematics , first published in Bell has a reputation as a highly inventive writer a trait not always considered a virtue in a biographer or historian. When it comes to the arithmetic, however, Bell is one of the few writers who scruple to distinguish between fact and conjecture.
It's a challenge to sort out patterns of influence and transmission in such a collection of stories. When the example given is , however, it's not so easy to trace the line of inheritance—if there is one. And the dozen or so other sequences that appear in the literature argue for a high rate of mutation; every one of those examples had to be invented at least once.
Tellers of a tale like this one seem to work under a special dispensation from the usual rules of history-writing. Authors who would not dare to alter a fact such as Gauss's place of birth or details of his mathematical proofs don't hesitate to embellish this anecdote, just to make it a better story.
They pick and choose from the materials available to them, taking what they need and leaving the rest—and if nothing at hand suits the purpose, then they invent!
For example, several authors show a familiarity with Bell's version of the story, quoting or borrowing distinctive phrases from it, but they decline to go along with Bell's choice of a series beginning , falling back instead on the old reliable or inserting something else entirely. Thus it appears that what is driving the evolution of this story is not just the accumulation of errors of transmission, as in the children's game "whisper down the lane"; authors are deliberately choosing to "improve" the story, to make it a better narrative.
For the most part, I would not criticize this practice. Effective storytelling is surely a legitimate goal, and outside of formal scholarly works, a bit of embroidery on the bare fabric of the plot does no harm. A case in point is the theme of "busywork" found in most recent tellings of the story including mine. The idea that he wanted to keep the kids quiet while he took a break is entirely a modern inference. It's probably wrong—at best it's unattested—and yet it answers a need of readers today.
An example of the latter position is the following account written in by three fifth-grade students, Ryan, Jordan and Matthew:. Am I being unfair in matching up Eric Temple Bell against three fifth-graders? Unfair to which party? Both offer interpretations that can't be supported by historical evidence, but Ryan, Jordan and Matthew are closer to the experience of classroom life.
As with the identity of the series, the details of how Gauss solved the problem remain a matter of conjecture. The algorithm that I suggested—folding the sequence in half, then adding the first and last elements, the second and next-to-last, etc. A related but subtly different algorithm is mentioned by many authors. The idea is to write down the series twice, once forward and once backward, and then add corresponding elements.
For the familiar series this procedure yields pairs of , for a total of 10,; then, since the original series was duplicated, we need to divide by 2, arriving at the correct answer 5, The advantage of this scheme is that it works the same whether the length of the sequence is odd or even, whereas the folding algorithm requires some fussy adjustments to deal with an odd-length series.
One way to envision this process is to fold the series in half with a hairpin bend. Another approach is to write the series twice, once in ascending and once in descending order. A third method selects just a single pair of elements, typically the first and last, in order the calculate the average. A third approach to the summation problem strikes me as better still. The root idea is that for any finite set of numbers, whether or not the numbers form an arithmetic progression, the sum is equal to the average of all the elements multiplied by the number of elements.
Thus if you know the average, you can easily find the sum. For most sets of numbers, this fact is not very useful, because the only way to calculate the average is first to calculate the sum and then divide by the number of elements. For an arithmetic progression, however, there is a shortcut: The average over the entire series is equal to the average of the first and last elements or the average of any other elements symmetrically arrayed around the midpoint.
You're so smart -- why don't you share your answer with the class? It was true. Gauss had figured it out In his head At 9 years old Do you hate him too? Unknown January 5, at AM. Unknown January 14, at PM. Unknown February 11, at PM. Unknown April 29, at AM. Unknown May 29, at AM. Unknown June 19, at PM. Unknown July 17, at AM. Unknown August 17, at AM.
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