- 30 Oct '02 11:49A rather bright but troublesome student was giving his math teacher a difficult time. To keep him

occupied for what she thought would be a lengthy period, she assigned him the problem of adding

together all the numbers from 1 to 100. Much to her dismay, he came up with the solution in a

matter of seconds. How did he do it? - 30 Oct '02 16:11remember what I said about me & maths earlier? This would be where

I showed off my amazing abilty to tap many numbers into a calculator

very quickly.

I can see how both of these answers are essentially the same, and

that they give the correct answer. Is it possible to explain to me why? - 30 Oct '02 16:59Jon,

The student was asked to add up the first 100 integers

1 + 2 + 4 + ... + 97 + 98 + 99 + 100

See what happens when you write the same sequence directly

underneath:-

1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100

100 + 99 + 98 + 97 + ... + 4 + 3 + 2 + 1

Instead of adding up the rows, add up each individual column. In

other words, 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101 and so on all

the way up to 99 + 2 = 101, and finally 100 + 1 = 101

Clearly, every "column" equals 101. The cunning bit I guess is that we

know how many columns there are. There must be 100 (since we've

written out 1, 2, 3, etc up to 100).

100 lots of 101 equals 10,100. So far so good. But why is the answer

5050? Remember that we added up the columns. Each column

contained 2 rows, that is we added up the first 100 integers *twice*.

Therefore we need to divide the answer by 2.

We had 100 lots of 101 and needed to divide by 2.

(100x101)/2 is the same (written slightly differently) as what Gilbert

and Hotpawn wrote.

Gauss realised all this in his head in a few seconds. The other children

would have got as far as 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 and so on,

adding each number in turn to the total of the previous numbers

added together. This is cumbersome and heavy handed.

As is so often in Mathematics, it wasn't that Gauss was brilliant at

adding up each number in turn, but rather he found an alternative,

simplified, more beautiful way of solving the problem.

Godfrey Hardy (might have been someone else) said "there is no

place for ugly mathematics".

Mark

The Squirrel Lover - 30 Oct '02 18:47Thanks Mark. As with most things, it's simple once you're shown how -

I knew where you were going as soon as you wrote the same

sequence directly underneath.

Is this a simple example of a useful and wide ranging mathematical

tool/principle or just an isolated (but pretty) phenominon? - 30 Oct '02 19:30It can be generalised to any arithmetic series (a sum in which the difference between one

term and the next is constant), so yes it is very useful. A similar technique can be used to

evaluate many other kinds of sums: you express them in the form

(a-b) + (b-c) + (c-d) + ..... + (y-z), and everything cancels except for a - z, which is your

answer. Nice. - 30 Oct '02 16:40It's a great story.

The student in question was Gauss. Historians disagree when he

actually did it but most concur that it was sometime between the ages

of 7 and 10 (probably aged 8) around about 1785. This paved the way

for the formalised theory of Arithmetic Progressions and Geometric

Grogressions.

I remember reading somewhere that what really transformed him from

precocious youngster to establised mathematician however came when

he was 19. He proved that a regular polygon of 17 sides could be

constructed with a compass and a straight edge. 2,000 years previous

the Greeks were obsessed about compass and straight edge

constructions.

Euclid had given constructions for regular polygons with 3, 4, 5, 6, 8,

10, 12, 16, 20, 32, 40, 48, 64 (and so on) but 17 proved quite tricky

for 2,000 years until Gauss came along (with help from work done by

Fermat).

For anyone interested in reading about Gauss then a good place to

look at is

http://www.geocities.com/RainForest/Vines/2977/gauss/gauss.html