Thursday, 28 July 2016

Take The Gauss Challenge!

All of the really clever stuff behind digital audio is just mathematics, pure and simple.  I’ve said before - and will say again - that mathematics is the purest of the pure sciences, and for that reason is often separated out from the rest of science.  The principal differentiator is that in the pure sciences the ultimate standard of correctness is measured in terms of empirical evidence, whereas in mathematics it lies in absolute proof.  Also - somewhat more esoteric in concept - scientific postulates usually start with a simple everyday observation, and proceed from there to both break it down into its component constructs and expand it into more elaborate implications.  By comparison, mathematics begins at ground zero.  It starts by defining the quantities ZERO and ONE, plus the concept of ADDITION, and builds upwards from there.

Mathematics, like Physics, also has the capacity to lead us down some mind-bending rabbit-holes.  One such example is Kurt Gödel’s famous Incompleteness Theorem.  Among other things, this theorem [a theorem is an axiom which has been proven mathematically to be correct - as opposed to a theory, which is an unproven postulate] states that there are some things that are mathematically correct, but which are fundamentally incapable of being proven.  Furthermore, he shows that it is impossible to prove that any given unprovable theorem is indeed unprovable.  But we don’t need to go there ….

I should observe that while I talk about mathematics in such glowing terms, I myself am a physicist.  I was equally qualified to study physics or mathematics at university (music too, for that matter), and wisely chose physics.  While at university I could keep up with my two mathematician friends for most of my first year, but by the second year it became obvious that what they were studying went way over my head.  Waaaaay over.  Good call.

One of the greatest mathematicians ever was a German named Carl Friedrich Gauss.  A famous true story is told of him, one against which you might be interested to try to measure yourself.  There is a nice, simple formula that gives you the sum of all integer numbers (1 + 2 + 3 + 4 + 5 +, … etc) up to any arbitrary number N.  That formula is N*(N+1)/2.  My challenge to you is to prove it.

The proof is erroneously attributed to Gauss.  As a 10-year old boy attending school in 1787, Gauss’s class was asked by their teacher to add up all the whole numbers between 1 and 100.  This gave the teacher some spare time to take a nap.  However, after ten minutes Gauss woke him up with the correct answer - 5,050.  He couldn’t be bothered to do all the adding up, and so had derived the above formula on the spot.  I don’t know about you, but I had not even been introduced to algebra at age 10!

In actual historical fact, the teacher - one Herr Buttner - was not looking to take a nap, but wanted to prepare his pupils’ mindset so that when he subsequently presented them with the formula they would better appreciate its worth and value.  A good teacher, I think!  Astonished by young Gauss’s precocious genius, he became his personal champion and was largely responsible for much of the young man’s early advancement.

The proof itself is in fact a deceptively simple one, which almost anyone could understand, and easily teach to others.

So I won’t bother to repeat it here :)