Can I Count on You?

Let's start with something simple, like counting. One, two,
three, four, five ... not so hard is it. Suppose we start adding
these numbers together, as follows:

1 = 1

1 + 2 = 3

1 + 2 + 3 = 6

1 + 2 + 3 + 4 = 10

1 + 2 + 3 + 4 + 5 = 15

Nothing too profound here, but suppose I asked you for the
sum of the first 100 numbers. In other words the sum of 1 + 2 + 3 +
(keep going from 4 to 98) + 99 + 100. (In the future I'll write ...
instead of (keep going from 4 to 98.)) This was an arithmetic
problem for young Carl Friedrich Gauss, the most famous
mathematician of all time, when he was in first grade. He
immediately responded with 5,050 the correct answer. Now this was
in the days before pocket calculators, even though that probably
wouldn't help you much anyway. I usually make a mistake after
typing in 3 numbers into a calculator, imagine trying to add up a
hundred numbers. So how did he do it? We all know that 2 + 3 = 3 +
2 right? When you add a bunch of numbers together, the order in
which you add them doesn't matter. Well, he noticed that if you
rearrange the sum as follows 1 + 100 + 2 + 99 + 3 + 98 + 4 + 97
+... + 50 + 51, that every two terms add up to 101, and there are
50 of them. Check it out, 1+100 = 101, 2+99 = 101, and 3+98=101.
Since there were 100 numbers to start with, there are now 50 =
100/2 pairs of numbers. So the total has to be 101 * 50 = 5,050.
Simple isn't it, but pretty clever for a six year old. Are you
still with me? All we have done so far is add up the same list of
numbers, but in a different order. Now lets go a little further.
Suppose we don't want to say exactly how many numbers we should add
up. How can you do that?

1 + 2 + 3 + 4 ... + (N-2) + (N-1) + N = ?

All I've done is explicitly show you the last few terms of the sum "symbolically." Instead of ending with 98 + 99 + 100 or 998 + 999 + 1000, I've used the letter N to represent the last number. Now let's apply the profound notion that 2 + 3 = 3 + 2 and rearrange the sequence like Gauss did. We get

(1 + N) + (2 + N-1) + (3 + N-2) + ...

We can write (2 + N - 1) as (N + 2 - 1) = (N+1). Also we can write (3 + N - 2) as (N + 3 - 2) = (N+1). Each of these terms adds up to N+1, and there are exactly N/2 of them, so what is the total? Well, it must be (N+1) times N/2, or if we write it the way mathematicians do, N*(N+1)/2 Now plug a number for N into this expression, and we have the sum of the first N numbers. If N is 5 you get 5 * 6 / 2 = 15 correct? If N is 100 you get 100*101/2 = 50*101 = 5,050 just like Gauss. It is a lot easier to calculate N*(N+1)/2 than it is to add up N numbers, yet the answer is always the same. This idea of letting letters stand for numbers is the heart of algebra, and allows us to express very complicated ideas in a very simple way. Now why didn't they tell you it was that easy in high school?

Quote of the day:

*When your wife leaves you
for your best friend, no one knows who is the lucky man.*

**An old Russian saying**

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