The Allure of Transcendental Numbers

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Deckard
 
Reply Sun 16 May, 2010 01:15 pm
@ughaibu,
ughaibu;164872 wrote:
Hippasus of Metapontum is the guy in the maths martyr stories. Sometimes he's killed for proving irrationality, sometimes for revealing it to non-Pythagoreans. Usually, he's drowned at sea.

For a split second I thought there was a graphic novel series called the Math Martyr Stories. That's just crazy enough I might have to make it myself.
 
Reconstructo
 
Reply Sun 16 May, 2010 10:49 pm
@Deckard,
Deckard;164821 wrote:
I wonder if nth decimal place of Pi or e or phi could be approached with probabilities. Could some probabilistic theory predict though not exactly what the nth decimal place is more likely to be?


Check this out. It's amazing. You might want to click on that png link to see this strange equation. Nice that it works in hex or binary as if custom tailor for the computers that would process it.
Quote:

An important recent development was the Bailey-Borwein-Plouffe formula (BBP formula), discovered by Simon Plouffe and named after the authors of the paper in which the formula was first published, David H. Bailey, Peter Borwein, and Simon Plouffe.[51] The formula,
http://upload.wikimedia.org/math/0/3/7/037bcc5ddc36d7cb44f83b6c5365027f.png is remarkable because it allows extracting any individual hexadecimal or binary digit of π without calculating all the preceding ones.[51] Between 1998 and 2000, the distributed computing project PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth (1,000,000,000,000,000:th) bit of π, which turned out to be 0.[52]


---------- Post added 05-16-2010 at 11:52 PM ----------

Deckard;164821 wrote:

With any given preceding digit x there are 10 possibilities for what digit comes next. Is the probability for each of these 10 possibility always 1 in 10? As the computers give us more decimal places do we approach this 1 and 10 probability?

I think pi is considered a "normal" number, meaning that the probabilty remains 1 in 10, as the ten digits are present in equal proportions. Normal number - Wikipedia, the free encyclopedia
 
ughaibu
 
Reply Mon 17 May, 2010 03:01 am
@Reconstructo,
Reconstructo;165178 wrote:
I think pi is considered a "normal" number, meaning that the probabilty remains 1 in 10, as the ten digits are present in equal proportions.
However, this conjecture has yet to be proved. On the other hand, Calude has proved that all random reals are Borel normal.
 
Zetetic11235
 
Reply Tue 18 May, 2010 01:44 pm
@kennethamy,
kennethamy;164848 wrote:
I hope not!................Anyway, for many, love and passion substitute for knowledge and understanding. In fact, many cannot tell the difference.


I had always thought that love and passion fuel our desire to understand and gain knowledge, that and necessity and anxiety and fear and curiosity.

I don't know what you mean by 'substitute' in this context. Substituting love and passion for knowledge and understanding; I can only assume you mean the sophomore who begins to think of his area of learning as mystical and all-encompassing such as the new student who thinks that all life is physics or chemistry or theater etc.

On topic; I think that there is much more to the transcendental number than there is to rational numbers. The transcendental numbers, to my knowledge, first came about from measurements. So we had assumed that the rational numbers would suffice for every measurement, but we found a 'natural' constant, pi. We took our artificial idea of a perfect circle (which is a syntactic approximation of a class of real objects, a way to pass from objects to computations about the objects, I suppose it is an abstraction of a very specific type) and we imposed our system of measurement on it. It did not have a representation; no matter what rational manipulation we carried out we either overshot it or undershot it. Hence the eventual infinite series representation, an algorithm that is guaranteed to bring you closer and closer to the value of the number (and not only closer, but they tell you exactly how many decimal places it is accurate to if you know the error bound formulas).

The main thing that transcendental numbers advised us of was that our concept of number was incomplete with respect to what we wished to use if for. We needed additional mechanics to describe these numbers; and this was achieved in the calculus. Eventually it was given a deeper level of logical structure by Bolzano, Weistrauss, Dedekind, Cantor (indirectly), among others.

In non-standard analysis which sought to realize Leibnitz's intuition for the calculus, we think of every number as an infinite sequence. The mechanics for realizing this intuition were very powerful and came from modern logic as it was in the 1940's. Abraham Robinson was the man responsible for this vindication of Leibniz. The intuition, very roughly, is that a number is a class of infinite sequences of numbers which only disagree at a finite number of places, so if we have, for instance; [1,2,3,4,5,6,7,7,7,8,8,8,8,8,8,8.....8,....8...] and [4,7,6,9,8,0,1,23, 57, 8,8,8,8,8,8,8......8.....8.....] these are considered part of the same 'equivalence class' with the name 8* representing the number 8. However, there are other numbers considered such as [3.1, 3.14, 3.1415.....] which are transcendental and then there are the infinite and infinitesimal numbers such as [1,2,3,4,5,6,7,8,9,10.....] and [1, 1/2, 1/4, 1/16....] which respectively exceed or decrease past any finite equivalence class.

The construction is based on the set theoretic notion of an ultrafilter. If you PM me I can give you some more information on that.

This is very loose and not at all a totally sound representation of Non-standard analysis. One last note; non-standard analysis is logically equivalent to real analysis or 'rigorous calculus' so any first order logical sentence that holds under a non-standard model holds in real analysis. However; statements about things like the least upper bound etc. that have second order definitions do not transfer into non-standard analysis so other tools have to be used there (tools which do not transfer back into real analysis).
 
Reconstructo
 
Reply Tue 18 May, 2010 02:26 pm
@Zetetic11235,
Zetetic11235;165830 wrote:
We took our artificial idea of a perfect circle (which is a syntactic approximation of a class of real objects, a way to pass from objects to computations about the objects, I suppose it is an abstraction of a very specific type) and we imposed our system of measurement on it. It did not have a representation; no matter what rational manipulation we carried out we either overshot it or undershot it. Hence the eventual infinite series representation, an algorithm that is guaranteed to bring you closer and closer to the value of the number (and not only closer, but they tell you exactly how many decimal places it is accurate to if you know the error bound formulas).

It was exactly this realization that brought me to the significance of math. I was thinking of Kantian and Hegelian categories. Blammo! Our spatial Euclidean continuity intuition does not compute in terms of our discrete sense of quantity. Of course the infinitesimal in early calculus was also a strange bird, and perhaps still is despite the sophisticated modern treatment of it. I can tell that you have more mathematical education than I, so I am curious as to what you can tell me concerning these things.

I must say that it did occur to me that transcendental numbers were algorithms. But I was helped by my amateur programming background in this case.

---------- Post added 05-18-2010 at 03:32 PM ----------

Zetetic11235;165830 wrote:
I had always thought that love and passion fuel our desire to understand and gain knowledge, that and necessity and anxiety and fear and curiosity.

Well said. And isn't fear just the inverse of desire? That which threatens the object of our desire? Hegel thought the desire for recognition drove history. And the desire for self-recognition drove philosophy. I feel that these numbers are part of us, that the real is rational and that man does not exist in isolation from the real. These numbers and the way we use them reveal some of the structure of the man-environment totality. Isn't it strange how eager some are to deny the passion of the mind? Is this not still the ugly side of religion? Is it not a terrible superstition that truth has nothing to do with passion? Why are we so eager, some of us, to reduce man to hunger and vanity? Philosophy, science, art, religion, mathematics...they all seem driven by "spirit." A sort of symbolic "instinct" --a program, to use a more modern metaphor.

---------- Post added 05-18-2010 at 03:35 PM ----------

Zetetic11235;165830 wrote:
I can only assume you mean the sophomore who begins to think of his area of learning as mystical and all-encompassing such as the new student who thinks that all life is physics or chemistry or theater etc.

Good point. We humans do tend to go a little crazy with our toys. On the flip side we hyper-analyze and negate one another is sophisticated in self-flattering ways. There's a joke among some friends of mine about the crazy college girls who obsess about psychology. Before long they couch their paranoid fantasies in this jargon that is ultimately as metaphorical as all the non-science it intends to replace. The worst superstition these says is half-science --an addiction to jargon. And yet jargon has its uses. And we must work with the language of the tribe.
 
 

 
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