Contemplating infinities.

Mathematics has many surprises - example a tad over 30% of all positive integers start with the first digit 1, whereas only 4.5% start with 9 - regardless of the fact the set is infinite. It just requires the set to be a base 10 number system - no more. Very counter intuitive - one would think there is a 1/9 chance - say 11% for any digit preponderance.

A phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon. Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability , much greater than the expected 11.1% (i.e., one digit out of 9). Benford's law can be observed, for instance, by examining tables of logarithms and noting that the first pages are much more worn and smudged than later pages (Newcomb 1881). While Benford's law unquestionably applies to many situations in the real world, a satisfactory explanation has been given only recently through the work of Hill (1996).

Benford's law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit is 1 almost one third of the time, and larger numbers occur as the leading digit with less and less frequency as they grow in magnitude, to the point that 9 is the first digit less than one time in twenty. This is based on the observation that real-world measurements are generally distributed logarithmically, thus the logarithm of a set of real-world measurements is generally distributed uniformly.

This counter-intuitive result applies to a wide variety of figures, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). The result holds regardless of the base in which the numbers are expressed, although the exact proportions of course change.

"You can't use it to improve your chances in a lottery," Dr. Nigrini said. "In a lottery someone simply pulls a series of balls out of a jar, or something like that. The balls are not really numbers; they are labeled with numbers, but they could just as easily be labeled with the names of animals. The numbers they represent are uniformly distributed, every number has an equal chance, and Benford's Law does not apply to uniform distributions."

In other words, Benford's Law needs data that are neither totally random nor overly constrained, but rather lie somewhere in between. These data can be wide ranging, and are typically the result of several processes, with many influences. For example, the populations in towns and cities can range from tens or hundreds to thousands or millions, and are affected by a huge range of factors.

Benford's law doesn't work for numbers controlled to a specific value, nor does it work for truly random numbers such as those generated by a random number generator. Benford's law also doesn't work well for small sample sizes. However, it holds true in a surprising number of situations. Benford's law shows that natural processes can be remarkably resistant to complete randomness.

Results from a "random" sampling don't prove that "a tad over 30% of all positive integers start with the first digit 1."

I wasn't attempting a proof with a sample - I was looking for a numerically significant inclination! The proof had already been tabled in number theory - I just couldn't absorb it without some real world validation.

An once you hit a few billion extremely large sample space datasets that support your theory - you take serious notice. I didn't ask at the time if the divergent / convergent bucket sized counting proof was accepted world-wide as an exhaustive proof (if only a power series converges is that necessary and sufficient to call the result). I would challenge that totally random "numbers" (versus random "data" - which may or may not be numbers) don't align to Benford's law - because one of Australia's top maths professors gave me that number generator and he was rather insistent that this was very slow - but the absolute best way to generate numbers that are as random as possible (due to the nature of occurrences of primes and how they generate the entire set of integers and give them their properties - a far more advanced field of number theory). One test doesn't show this is conclusive - but it stayed in my mind - do you have any information about what random numbers (say integers) don't comply with Benford's law? I agree - random colours or emotions wouldn't follow Benford's law - but colour or some data aren't inherently integer numerics so they aren't even up for consideration!

But when I say all positive integers - I meant all, not a subset - because some special subsets will not converge, but they tend to be the marked case (e.g. peoples height). When you say but Gambling doesn't show this trend - you have to recall that 1) some gambling has a restricted field - so this will effect your win results in a way that may or may not be a Poission or binomial distribution and or 2) the effort that goes in to making pseudo random number generators for gambling is enormous - and don't presume they generate numbers that are normally distributed either!

Once someone says "all positive integers" you are exposed to the behaviours of the entire group - an infinite group - that from human perception kinda starts at zero and expands outwards, not a finite slice of the group that human perception usually deals with.

You might find the proof I mentioned if you google more broadly into number theory rather than just Benford theory. I'll see if I can dig up my lecturer's e-mail and shoot her a line to ask specifically when mathematicians had the first accepted proof of Benford's law - cause I accepted it was alot before the late nineties.

You can't say there are the same number of integers and squares - becuase same is a mathematical instruction for equals - and equals is a operation on a number - not an infinity.