Natural Numbers - Pi, Log, ....

Chaos' lil bro Order

Greetings,

I'm far from an expert on math and I wanted to appeal to smarter minds to help compile a list of numbers with non-repeating chaotic decimals:

Here's the first 2 (the only 2 I can think of)

Pi
e

I'd also include the order of prime numbers because although it can be approximated as to when the next prime will occur, it is still random and (currently) non-deterministic.

Can you add some numbers or sequences that fit this mold?
Thanks.

chroot

log is not a number, it's a function. Perhaps you mean 'e.'

The square root of two and three and so on are also irrational numbers. There are, in fact, an infinite number of irrational numbers.

- Warren

HallsofIvy

As chroot said, there are an infinite number of irrational numbers and all irrational numbers are "non-repeating" (I don't know what you mean by "chaotic" here) decimals. In fact, in a very specific sense, "almost all" numbers are irrational. And, of course, the set of irrational numbers is uncountable so there cannot exist a "list" of them!

selfAdjoint

There are power-of-the-continuum transcendental numbers (that is, they can be put into one-to-one coreespondence with points on the line). They are defined by not being the root of any algebraic equantion (irrational numbers are just not the root of any linear equations). Because any recursive pattern in the decimal expansion would, I would think, set up an polynomial of which the number would be a root, I don't think there is any finitely generate recursive patter in the decimal expansion of a transcendental number.

e and pi are the two best known transcendental numbers. e was shown to be transendental by Hermite, the same man who is known for Hermitian operators (fun fact: his high school math teacher also taught Galois). Pi was shown to be transcendental by Lindemann in the 1890s.

Gel'fond proved the beautiful theorem that if b is any transcendental number and a is any algebraic number (i.e. not transcendental), then b^a is transcendental.

Chaos' lil bro Order

Thanks guys, very nice answers.

My question was poorly worded, but your answers were great.

kesh

finite or infinite, a complete list is impossible. cantor proved that such a list cannot be created, even if it were to continue without end

look into cantor's diagonal argument. it's pretty accesible even to non-experts

only rational numbers have repeating decimals. irrationals like sqrt 2 don't repeat.

no-one has proved if pi, e, sqrt 2 have evenly distributed digits, equivalent to 'random' (that is are normal numbers), it's just a hypothesis

what i love about normal numbers is that if you represent them in base say 30 - using the alphabet, a space, punctuation, to represent the 30 possible digits - then somewhere in the 'digits' must appear any message you'd like. the bible, shakespear, etc.

seeing as pi is 'probably' a normal number, somewhere in it's digits is the message "pi is exactly twenty-two over seven, dumbass"

monkey, typewriter, infinite time etc.

Chaos' lil bro Order

You raise an interesting point. Since pi has infinite non-repeating decimals (as far as we know) all messages are contained within it. In fact, every letter of this post is embedded in it somewhere. Cool idea, even thought its completely impractical.

AlphaNumeric

It's not 'as far as we know' (that implies we're not certain about it), it's proved because pi is proved irrational.
That is true of any truely random infinite series of numbers and any method of converting those numbers into letters.

However, that doesn't mean it's true just because it's irrational. For instance, the first number to be proved transcendental (and all transcendental numbers are irrational) was

[tex]\sum_{n=1}^{\infty}10^{-(n!)}[/tex]

I can, with 100% certainty, tell you that the digit '2' never appears in that number's decimal expansion, yet it is infinite and non-repeating. It's another one of those 'quirks' of infinitly large sets.

For instance {2,3,4,....} is infinitely large but misses out the number 1 (to say nothing of negative numbers, other rationals, irrationals and transcendentals!) yet never repeats a number as you move through the list.

CRGreathouse

I think "as far as we know" was intended to apply to "are", not "has". We know pi has an infinite non-repeating decimal expansion, but we don't know that every sequence is contained in it. (This is true for all numbers normal to base 10, but pi hasn't been proved normal to any base...)