Friday, September 10, 2010

Rant: We Count Wrong

We count wrong.
OK, so integral base-10 works pretty well for most human endeavors, and floating-point base-10 covers most of the rest.
Kinda like Newtonian physics: good enough for small orders of magnitude.
Then there’s π.
Here’s a fundamental mathematical constant which is simple and ... untidy.
For all practical purposes it’s a perfect random-number generator – it’s that untidy.
We are amazed at the “transcendental” nature of such a simple and fundamental ratio.
We pick some arbitrary number of digits and make do with that approximation.
We try calculating it, and are amazed at the elegance of the patterns within the equations.
We obsess over how “cool” π is.
We’ve got it backwards.
It’s not that π has an infinite number of digits seemingly random yet produced by elegant equations.
It’s that our number system is grossly inefficient.
It’s not that π is irrational.
It’s that we are irrational.
We count wrong.
e = 2.71828182845904523536…. in base-10.
Like π, e is transcendental and fundamental, with many of the same characteristics.
Many years ago I stumbled across someone’s observation that e – akin to π – could be expressed in “base-factorial” notation in a very clean way. Rather than each digit being a simple order-of-magnitude multiplication, it represents a factorial multiplication. Follow the link for more confusionclarification.
e = 10.011111... in base-factorial.
Expressed in the right terminology, it’s very simple.
π exhibits much of the same behavior, lacking only suitable simple expression.
One of the great failings of humans is the insistence on forcing everything to fit within our prejudices.
Great success oft comes from getting over those prejudices and accepting what is as it is.
We have a profound ingrained prejudice born of our DNA-influenced number of extremities.
Methinks the great hindrance to human mathematical progress is our stubborn insistence on mapping the universe to our fingers.
This works fine for small orders of magnitude – just like Newtonian gravity works fine for falling apples.
This gets untenable for large orders of magnitude – just like Newtonian gravity doesn’t work for apples falling at speeds near that of light.
Approximations to a few decimal places work fine for most cases.
To be correct, however, we find the theory of base-10 counting just doesn’t work.
Computing π to any significant degree takes enormous amounts of effort; figuring any given digit requires figuring all the digits before it.
When we took a change in counting seriously, by embracing and following base-2, we changed the nature of human knowledge in a few years.
Computing π in base-2 is, in fact, easy; figuring any given bit can be calculated directly, independent of bits before and after.
Our counting, and by extension our math, is wrong.
We count wrong.
Computers demonstrate it.
π proves it.
I now return you to wondering what the he11 I just wrote and why.

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