Gauss` Summation 1+2+3 Anecdote

Summation, adding up of sequences, sigma summation operator – these terms send shivers down the spine of many algebra students. Surprisingly, these are also the tools of one of the most comical and math-morale boosting anecdotes in the history mathematics.

The German mathematician Carl Friedrich Gauss (1777 – 1855) was a genius even in childhood. He attended primary school in Brunswick (Braunschweig) in the Duchy of Brunswick-Wolfenbütte in Germany. At the age of 10 the young Gauss was admitted to the algebra class of math teacher J.G. Büttner. To keep the bored and unruly schoolboy Carl Gauss busy for a good long time Büttner gave Gauss the following task: add-up the integers 1 to 100 in arithmetic progression. As Büttner turned back to teaching his class , the young Gauss placed his slate on the table, saying, “There it is!” Büttner looked back at him scornfully. To Büttner it must have been utterly impossible that a boy, even as talented as Gauss, could produce an answer within seconds. But when Büttner finally looked at the results on Gauss’ slate and it was he who probably felt cold shivers going down his spine: On Gauss’ slate was written the correct answer, 5050. No further calculation, such as 1+2+3+4+… was written, just the right answer: 5050. Büttner was utterly astonished and puzzled. He even called his assistant Martin Bartels. They asked the young boy how he possibly can come up with the right answer so quickly.

As Gauss explained, he realized that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050. In more algebraic terms, Gauss had found the explicit solution to adding up the numerical sequence from 1 to 100. The explicit summation formula looks like this:

By using mathematical induction it can be shown that the sum of all natural numbers n equals the sum of the first (1) and last integer (n) multiplied by half-of-the sequence elements (n/2) which yields the correct answer. It was totally amazing that this 10 year old boy came up with this method in his head within seconds.

Impressed by Gauss’s intellectual abilities Büttner presented him to Duke of Brunswick, who sent him to what is today the Braunschweig University of Technology.

The importance of the math genius Gauss to Germany is shown in part by the fact that Gauss’s portrait is shown on German Bundesbank’s ten-Mark banknote issued from 1989 through 2001. The ten-Deutsche Mark banknote also shows prominent buildings in Göttingen (the town  Gauss worked for many years), and a normal distribution curve — one of Gauss famous mathematical discoveries.

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Gauss impressed his algebra teachers so much that he was awarded a scholarship. Perhaps this article inspires you to do the same!

Editor’s footnote: Being a German and having lived at the time of issuance of the Gauss ten-mark bill I not only enjoyed the beautiful styled banknote but also its purchasing power. The Gauss banknote felt like real money, in sharp contrast to the ever depreciating value of today’s ten euro banknote.