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© Copyright 1997, Jim Loy
[Imagine that you are in the audience at this lecture]
There is a story about Carl Friedrich Gauss. Supposedly, when he was a little boy, his teacher asked the class to add up the numbers one through a hundred (1+2+3 etc., all the way up to 100). The teacher wanted to get some work done, or get some sleep, or whatever. Anyway, to the teacher's annoyance, little Gauss [Here the lecturer holds his hand out to show that little Gauss was about 2 feet tall, to the amusement of the audience]... To the teacher's annoyance, little Gauss came up to the teacher with the answer, right away. The teacher probably had to spend the rest of the class time verifying little Gauss's [2 feet tall] result.
Some people find that story hard to believe, even impossible. I think that the story has the ring of truth to it. I believe that the story is true, or close to it. There are versions of the story, in which the numbers are one to a thousand [murmur in the audience].
I think that you people can duplicate little Gauss's [2 feet tall] trick [doubt in the audience]. I'm going to give you two very small hints. But, that's all you will need, to be just like little Gauss [2 feet tall].
Nobody use your calculators, or even paper and pencil for a while. You are going to be slower than little Gauss [Lecturer hesitates, then shows "2 feet tall"]. But, you're going to be just as smart.
Here's the problem:
1 + 2 + 3 + ... + 99 + 100 = X
We want to find X. Well, it's going to take 99 additions to solve this. Let's see, 1+2=3, +3=6, +4=10, +. It's going to take a while. There's got to be an easier way.
What if we start at the other end:
100 + 99 + 98 + ... + 2 + 1 =
Do we get the same answer? [Various forms of "yes" from the audience] How many think we get the same number, raise your hands? [Most raise their hands] Yeah. It was algebra, right? Associative Law? It doesn't matter what order you add things up, you get the same answer. So "yes" we get the same answer [Lecturer writes "X" to the right of the equal sign].
That was your first hint, "Associative Law."
Let's see, 100+99=199, +98=297, +. That's going to take just as long, isn't it? There are 99 additions there, too. That didn't help. Any other way?
What if we add up the even numbers (that's 49 additions), then add up the odd numbers (that's 49 additions), and then add up the two totals? That's, uh, 99 additions. Darn, that's no better. How about this?
1 + 2 + 3 + 4 + ... + 48 + 49 + 50 51 + 52 + 53 + 54 + ... + 98 + 99 + 100
When we finally total them up, we get the same answer, right? Associative Law? Does that look helpful? Not [hesitate] really.
1 + 2 + 3 + 4 + ... + 48 + 49 + 50 100 + 99 + 98 + 97 + ... + 53 + 52 + 51
Does that help? This is the second hint, by the way [points at those numbers]. Do you see something magical about that? Anybody see it?
[A couple people in the audience say that the columns all add up to 101].
[There is general agreement].
Right. Do you all see it? [The lecturer points at each column in return] 101, 101, 101, ..., 101, 101.
How many 101s do we have? [Pause. Some say "50." There is agreement] Right. 50 times 101:
50 x 101
Can we do this in our heads, or do we have to use our calculators for that?
[Some people say "5050"].
[The lecturer writes a big 5050 on the board, and begins applauding the audience].
[The audience applauds].
You see that it is not much of a leap from the Associative Law to the answer. You just have to figure out a useful way to arrange the numbers.
And, adding up the numbers from one to a thousand is no more difficult than one to a hundred. It's the same problem. There are just 500 columns, each of which add up to 1001. 500 x 1001:
The sum of the numbers from one to one hundred form what is known as an arithmetic series. How about the sum of an arithmetic series in general. Such a series may look like this:
In this series the first element (a) is 7, the common difference (d) is 4, and the number of elements (n) is 7. In our Little Gauss story, a=1, d=1, and n=100. Here is the general sum:
We have seen something like that last series (T=1+2+3+...+(n-1)) already, in our story. And we know how to handle it (pretend that n-1 is even):
1 + 2 + 3 + ... (n-1) + (n-2) + (n-3) + ... T = n + n + n + ...
And how many n's are there? Why there are (n-1)/2 of them. So T=n(n-1)/2. And S=na+nd(n-1)/2. We get the same formula when n-1 is odd. Just go from 0 to n-1, making it an even number of elements. There are other ways to derive the same formula. And there are different forms of the same formula.
So, use our formula on Gauss' 1 through 100. S=100+100(99)/2=5050, which is what we got above.
Incidentally, Gauss is spelled Gauß in German. It is similar to Johann Strauß.
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