e, pi and phi are irrational numbers, and they are interesting because they are expressing proportions that can be found in many, so called, different systems.

If some proportion is found in many systems, we hope to find through it if there is some deep connection between these systems.

Shortly speaking, we are talking about the signature of some deep symmetry that can be used as a gate between, so called, different systems.

If we find some deep symmetry between, so called, different systems, then this deep point of view, gives us the opportunity to explore these systems from deeper and higher level of understanding.

Because e, pi and phi are irrational numbers, I think we have to start our research by asking ourselves: "what is an irrational number"?

A better answer to this important question can give us a deeper understanding of the connections between e, pi and phi.

By standard Math, irrational number is a number that cannot be expressed by a ratio that exists between at least two integers.

If this is the case, then we have no accurate method to represent an irrational number.

Can somebody have an idea how to represent an irrational number in an accurate way without using the natural numbers notations?

Please be aware that notations like e, pi or phi or pi/e are general notations exactly like oo is for infinity, because they do not give us any accurate but only a trivial information about these irrational numbers.

Another way to think about this problem, is to agree with the idea that redundancy_AND_uncertainty are natural properties of the NUMBER concept right from the level of the natural numbers, for example:

Any of these numbers can be represented with sufficient precision for any possible use. If you do not need digits of precision then the symbol will do nicely. I suppose you could create a number system where one or all of these numbers are integer or rational. Of course the payback would be that 1 would be irrational.

Notice also the unjustified (unjustifiable?) presumption that writing two numbers as ratios of integers constitutes an exact representation, and is the only possible exact representation, whatever that might mean. Heck, why is even an integer an exact representation of a point in the real numbers?

if I am not mistaken, or at least for pi, their is not a single way to do so, since it is trancendent, this has been proven.... or perhaps I have misunderstood your question, what exacly do you meen with "the natural nimbers notations"?

All irrational numbers can be represented exactly by infinite series. Defining numbers without using numbers, now you are sounding like someone who has been puffing a bit too much on the old crack pipe.

ok. it's sqrt(2) or [tex]\sqrt2[/tex] or [tex]2^{1/2}[/tex] they're all exact 'notations' of that quantity which is the unique positive root of x^2-2

Sorry that you think decimals ARE real numbers, and that terminating decimals constitute the only exact things, which is a not a good way of thinking of these things. I suppose we can understand the idea that only fractions are nice, but this presupposes that it is necessary to talk about the real line as if it were actually physically a line with little notches on it like a ruler. Seeing as you cannot mark any points on a ruler with any certainty I am at a loss to understand where you're comgin from.

They aren't decimals, or rather that is not their defining property, and as you well know the issues of thinking they are, one wonders why you persist in this view.

Erm, ok. as the only thing that defines it is it is positive and squares to two, let me think.... er, yep, it's an accurate notation as far as i can tell. what data behind it? that makes no sense, but you've already adequately demonstrated that you do not accept that the real numbers are cauchy sequences of rational numbers modulo the obvious equivalence, as it 'rapes' something, which is a bizarre choice of words.

Cauchy sequences of rationals using rational notations that some of them are finite therefore accurate, some of them has periodic returns therefore they are accurate by periodic returns.

Irrational numbers like sqrt(2) don’t have any of these properties, so by what property they can be accurate?

But you're putting your own particular spin on "accurate" which means: can be specified by a finite number of integers picked from the set 0,1,...,9 with possibly some indication of a recurrence. Fine, but that doesn't stop sqrt(2) being a perfectly well defined real number. I would dismiss your preference for accuracy like this as relatively unimportant. Constructively all numbers are equally hard to indicate on a ruler, and algebraically/analytically you're out of your depth already

If you knew about dedekind cuts you wouldn't make such statements about placing things on the real line (you speak as if it were a phyiscal line still), and is this the same meaning in accuracy as in the post before?

By your question we can see that you don't understand the meaning of 'not accurate', which is not the difference between accurate things but the self property of an element to be not accurate.