display | more...

(An entry in The Lives of the Mathematicians)

An old (probably false) story about Gauss tells how, at age 6, Gauss' schoolteacher wanted some peace and quiet, so he gave the class an exercise: add up the numbers 1 to 100. Naturally, he assumed this would take them a while.

Unfortunately, Gauss immediately reasoned as follows, and gave the right answer: pair off the biggest and smallest numbers, to get (1+100) + (2+99) + .... Now note that each pair adds up to 101, and that there are exactly 50 such pairs. So the sum is 5050.

The story how Gauss quickly added up the numbers 1 to 100 (100% fake story) is probably better, if even more groundless.


IMPORTANT WARNING! EVERYTHING BELOW THIS LINE WAS WRITTEN BY ME, BUT UNDER A SEPARATE TITLE. IT WAS MOVED BY SOME EDITOR (I USE THE TITLE GIVEN BY THE E2 GODS) AND DUMPED BELOW WHEN E DECIDED TO NUKE THE ORIGINAL. EVERYTHING2 BELIEVES IN YOU, YOUR WORDS, AND YOUR TITLES. EXCEPT WHEN IT DECIDES TO CHANGE SOMETHING. HAVE A NICE DAY.


Unlike the story of how Gauss quickly added up the numbers 1 to 100, which is only probably fake, this story was made up (I don't know whom by), so it is guaranteed 100% fake.

When Gauss was 6, his schoolmaster, who wanted some peace and quiet, asked the class to add up the numbers 1 to 100.

"Class," he said, coughing slightly, "I'm going to ask you to perform a prodigious feat of arithmetic. I'd like you all to add up all the numbers from 1 to 100, without making any errors."

"You!" he shouted, pointing at little Gauss, "How would you estimate your chances of succeeding at this task?"

"Fifty-fifty, sir," stammered little Gauss, "no more..."

The story of Gauss adding numbers together quickly is an important, if apocryphal, mathematical lore. Perhaps because it is apocryphal, perhaps because it involves the simplest of arithmetic operations, additions, it is often treated as nothing more than a parlor trick. But there is some interesting math behind it.

Lets generalize the results away from 1 to 100. In general, if our highest number is x, than each pair will add to x+1. And since we have half as many pairs as our x, we will multiple (x+1)(x/2). This gives us (x^2+x)/2. So for any given number, if we want to calculate the sum of its digits, we just square it, add x, and divide by two. Not that this is a particularly easy trick itself, squaring, say, 537 in our heads is not something we can do automatically. But it is certainly easier than the alternative.

There are a few interesting things about the expression (x^2+x)/2. First, as the number gets higher, the sum effectively becomes x^2/2. If you are interested in reciprocal sums, then, we know that since the denominator is squared, the sum has to converge. Since Leonhard Euler demonstrated that the sum of the reciprocal squares converges to pi squared divided by six, it might be guessed that this sum would converge to around 3. However, the early numbers make up an important part of its reciprocal sum, so it actually converged to...2. (I base this only on a spreadsheet of the first 10,000 or so terms, I have no mathematical proof or reasoning for why this would be.

The last paragraph might have been too densely mathematical for some, and too hastily reasoned for mathematicians, but the point is that something that seems like just a trick, that seems to have no wider application, can, using mathematical methods, be seen to fit a wider and more important pattern.

Note: after writing this, I have found that I have rediscovered "Triangular Numbers", and that these things are already known widely.

Log in or register to write something here or to contact authors.