On 23/01/2012 10:02 AM, John VanSickle wrote:

> The rigorous proof that all such series will have a member-to-member
> ratio approaching phi will entail showing that for any two non-negative
> integers, A and B, with B>A, that the value of B/(A+B) is closer to phi
> (1.618~) than the value A/B.
>
> |A/B - phi| > |B/(A+B) - phi| for all A and B, 0 < A < B
>
> The full proof is left as an exercise for the interested student.

I've always found an intuitive understanding to be more interesting than 
a rigorous proof.

For example, the square root of 2 is irrational. The standard way to 
prove this is to demonstrate that root 2 = A/B implies that A/B can be 
cancelled down an infinite number of times, which is clearly impossible. 
This shows that the number *is* irrational. But it gives no indication 
as to *why* it is impossible.

By contrast, root 2 = A/B implies that A^2/B^2 = 2. The only way this 
can be true is if A^2 is twice as large as B^2. And /that/ would mean 
that if you look at the prime factors of each number, A^2 would have one 
more 2 factor than B^2 does. However, the prime factors of X^2 are 
exactly the prime factors of X, but repeated twice. This entails that 
EVERY SQUARE NUMBER HAS AN _EVEN_ NUMBER OF PRIME FACTORS. But for A^2 
to have *one* extra factor of 2, either A^2 or B^2 would have to have an 
ODD number of 2 factors - which is impossible.

Here we clearly see /why/ you cannot make this equation work. We also 
see exactly which numbers can and cannot work in the equation. Namely, 
A^2/B^2 can be equal to any number having EVEN numbers of each prime 
factor. In other words, SQUARE NUMBERS. So the square root of any 
integer is either an integer or an irrational number. It can never be 
some non-integer rational.

Of course, from a rigorous point of view, proving that a fraction 
cancels down forever merely involves some trivial algebra, whereas 
proving the properties of unique factorisation and so forth is a major 
undertaking. I still find it much more intuitively illuminating.

Likewise, the Fibonacci numbers demonstrate how repeatedly applying a 
constraint to something causes it to approximate the unique solution to 
that constraint. That's a much more intuitive idea than any amount of 
fidgeting with equations.

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