Gauss and the Sum of Numbers from 1 to 100Date: 03/02/2006 at 14:00:25 From: Danielle Subject: Gauss's Trick I was wondering how Gauss's trick works. I know that he uses a variable to add up large sums of numbers like 1 - 100. I am trying to understand this trick as a future educator, and learn how to break it down so I can teach the concept to my students. I do not really know where to start and would appreciate any help. Thank you for your time. Date: 03/02/2006 at 15:32:32 From: Doctor Peterson Subject: Re: Gauss's Trick Hi, Danielle. I don't think we know just how Gauss himself thought through this problem as a child; we just know that he turned in his answer quickly enough to attract the attention of his teacher. (I've seen many variations of the story, probably embellished with details we can't be sure of.) It's worth noting, also, that the trick was not new to him, and was not a notable piece of math even at the time; it was just pretty good for a kid to come up with himself! There are several ways to do it; you can use algebra, or pictures, or averages, among other approaches. You can apply the formula for triangular numbers (which are sums of the form 1+2+3+...+n), which is a special case of the formula for summing an arithmetic series. Here's one method I like: Imagine adding up all the numbers: 1 + 2 + 3 + ... + 98 + 99 + 100 Now imagine writing the same sum in reverse: 100 + 99 + 98 + ... + 3 + 2 + 1 We can add up all 200 of these numbers (which will give us twice our sum) by pairing them off: 1 + 2 + 3 + ... + 98 + 99 + 100 100 + 99 + 98 + ... + 3 + 2 + 1 --------------------------------------- 101 + 101 + 101 + ... + 101 + 101 + 101 But that's just the sum of 100 of the same number, and is equal to 100 times 101. Our sum is half that, 50 times 101, and you can do that in your head: 5050. Here's the picture version of the same idea: o o o o o o o o o * o o o * * o o * * * o * * * * * * * * * Here I'm combining two copies of 1+2+3+4+5, so I can find the sum by taking half of a 5 by 6 rectangle. You can see here, by the way, why this is called a triangular number. Almost equivalently, you can just pair up numbers at opposite ends of the list: 1 + 100 2 + 99 3 + 98 ... 49 + 52 50 + 51 There are 50 pairs, each summing to 101, so the total is 5050. (This may well be what Gauss did, but I find it slightly more awkward than the first method, especially if there were an odd number to add up.) Using the idea of averages, you can just note that the average of all the numbers is obviously halfway between 1 and 100, namely 101/2; and the sum of 100 numbers with this average is equal to 100 times their average, or 100*101/2, which again is 50*101. I doubt that Gauss (or your students) would have used algebra, so I won't go into the algebraic versions. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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