The Genius of Gauss

With the royal wedding approaching on Friday, it seems fitting to quickly mention a man who was known as the 'Princeps mathematicorum', or the 'Prince of Mathematics'.  He also often referred to mathematics, as the 'Queen of Sciences'.

Carl Freidrich Gauss was an early 19th century mathematician and I have recently had the good fortune, to encounter one of his elegant mathematical feats.  I am referring to the anecdotal story of when he was a child at school.  His mathematics teacher asked the class to add up all of the integers from 1 to 100.  This task, by hand, would have taken some time, but Gauss is alleged to have pondered the problem momentarily - and then come up with an ingenious solution, to allow the sum to be done in a few seconds, using a simple formula.

He approached it, thus:

If you take the sum of the  numbers from 1 to 100 i.e.

1 + 2 + 3+ 4...+ 100

and make these equal to X

and if you then take the same numbers 1 to 100, but this time, you reverse them so you get:

100 + 99 + 98... + 1

These numbers added up, would also equal X.

Therefore, if you added together both of these series, they would equal 2X.

Easy enough so far.  But here is the really clever bit.  What Gauss did, was to add together, the two series of numbers in the order shown above.

For example he added:

(1+100) + (2+99) + (3+98).... + (100+1)

Now, all of these little sums such as 1+100, all add up to the same number, which is (101).  And if we say that in  our series from 1 to 100, that the number 100, is equal to (n)

Gauss realised that 101, is equal, in general terms, to (n+1), because each of the above terms such as (1+100), (2+99), is equal to (100 +1), or  (n+1).

He also realised that (n+1) must be multiplied by how many terms, he was adding up.  In this case, the terms from 1 to 100, equal 100 or (n) terms.

Therefore, he was then able to produce a general formula, to add together the sum of any sequence of consecutive numbers, which was:

n(n + 1) = 2X

This is simply, the sum of the two sequences of 1 to 100 and 100 to 1.  They are the same sequence, just in the opposite order.  So, they both still add up to 2X.

Therefore X = n(n+1)/2,  where X = the sum of the series of (n) consecutive integers.

So, in our above problem; Gauss wanted to add up the numbers from 1 to 100.  So, in this example, n=100.

Putting this into our equation, we find:

X = 100(100+1) / 2
X= 5050

If you do it longhand, and add the numbers from 1 to 100, it does indeed equal 5050.

Now, all this may seem a little complex; and, for a young child in a maths lesson, it was.  Yet Gauss, worked it out, in a very short period of time, and demonstrated his prowess and precocious nature, that was to have him crowned the prince of maths, studying the queen of sciences.

And not a single line of Union Flag bunting, in sight!