The Mysterious Connections Between Irrational Numbers - e, pi, and phi

[Organic]

do those constants have any relation to each other?

does something like pi-e or pi/e has any significance?

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:

http://www.geocities.com/complementarytheory/Complex.pdf

 

[matt grime]

That's not true is it. e, pi, phi are well known to be real numbers, and infinity isn't.

 

[Integral]

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.

 

[matt grime]

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?

 

[WWW]

But (by organic) maybe

redundancy_AND_uncertainty are natural properties of the NUMBER concept right from the level of the natural numbers

Is it possible?

 

[The_Imagizer]

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

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"?

 

[AntiMagicMan]

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.

 

[WWW]

But can an infinite series represented by integers notations can give an exact result of some irrational number?

If yes, then please write sqrt(2) for example.

 

[matt grime]

ok. it's sqrt(2) or \sqrt2 or 2^{1/2} 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.

 

[WWW]

Is the data behind sqrt(2) or 2^(1/2) notations is accurate?

Proove it.

 

[WWW]

Is the data behind sqrt(2) or 2^(1/2) notations is accurate?

Prove it.

 

[matt grime]

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.

 

[WWW]

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?

 

[matt grime]

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

 

[WWW]

What depth?

You show nothing but your belief that some irrational number has an accurate place on the real line.

 

[Hurkyl]

Let's reverse the question.

If you're so sure that sqrt(2) is not accurate, then please, tell us the difference between sqrt(2) and sqrt(2).

 

[matt grime]

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?

 

[WWW]

Dedekind using a Boolean Knife, which means he finds the property of its logical reasoning, which has no vagueness in it.

 

[WWW]

If you're so sure that sqrt(2) is not accurate, then please, tell us the difference between sqrt(2) and sqrt(2).

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.

It cannot be understood be false/true reasoning.

 

[matt grime]

then illuminate it for us and tell us what you think accurate means.

 

[WWW]

Accurate means a final state of information.

 

[HallsofIvy]

How about the belief that every irrational number has an exact position on the real number line. That is, after all, pretty much the definition of the real number line!

"You show nothing but your belief that some irrational number has an accurate place on the real line."

In other words, he is showing nothing but his knowledge of the real number system.

 

[PFanalog57]

x^2 + 1 = 0

x = sqrt[-1] = i

i^i = e^[-pi/2]


[1+i^i]*[1-i^i] = [e^pi - 1]/[e^pi]

A physical system is described by a normalized vector[state vector] in Hilbert space. All possible information can be known about the system, since, for every physical observable there corresponds a self adjoint operator in Hilbert space.

The only allowed physical results of measurements of some obervable U, are the elements of the spectrum of the operator which corresponds to U.

So all properties of a number may not be completely known, but that which is known, must be specifiable on logical or analytic grounds.

Now if you are trying to say that mathematics is inherently random at its foundations, you must define what randomness is ...exactly.

Take a coin toss for example, as the number of flips of the coin increase the
HTTTHTHTHHTHT...HTHTHTHTHT...

The probability becomes an "exact" number at an infinite limit. 1/2

So a number becomes an identity in the Platonic aeon.

 

[WWW]

In other words, he is showing nothing but his knowledge of the real number system.

His knowledge is based on the current paradigms of Math Language.

 

[WWW]

So a number becomes an identity in the Platonic aeon.

I prefer to think about simple symmetries that maybe can be found in the basis of, so called, different systems.

 

[Integral]

Using the iterative sequences.

S(n+1)= \frac {x_{n}} 2 + \frac 1 {x_n}

if we let x_1 = 1 we have a sequence which converges to root 2 from below.
If we let x_1=4 we have a second sequence which converges to root 2 from above.
Now I have constructed a set of nested intervals which have length going to zero. By theorems proven in Real Analysis, the intersection of these intervals contains a single Real number, in this case Root 2. Thus we have shown that an irrational number occupies a fixed well defined position on the real number line.

 

[WWW]

What is the meaning of the word 'converges' here?

Prove that above and below really reaching to a final state of information.

 

[PFanalog57]

What is the meaning of the word 'converges' here?

Prove that above and below really reaching to a final state of information.

lim
x->a f(x) = L

If 0 < |x - a| < delta then |f(x) - L| < epsilon

For example:

lim
x->3 (2x-5) = 1

if 0 < |x-3| < delta then |(2x-5)-1| < epsilon

|(2x-5)-1| = |2x-6| = |2(x-3)| = 2|x-3|

If 0 < |x-3| < epsilon/2 then |(2x-5)-1| = 2|x-3| < 2*epsilon/2 = epsilon


Aeon means eternal.

A timeless symmetry?

An infinite number of coin flips gives an equal amount of heads and an equal amount of tails.

[1/2 H and 1/2 T]*n, for n--->oo

A radioactive nucleus decays in accordance with probability P within time t_0 to time t_1

Probability P becomes a timeless mathematical entity governing the future iterations of events at time t. There exists a spectrum of possibilities for the observed quantities. Certain deterministic factors become contingent with respect to uncertainty, DxDp >= h .

An infinite number of observations of the radioactive decay, converges to an exact number for t?

Wave function probability density = |psi (r, t)|^2


The physical meaning of the expectation value appears to be simple. It is the value that would be found by taking the average of many measurements of the observables in question on a large collection of systems all in the state psi. the individual results are weighted by the probability.

 

[WWW]

lim
x->a f(x) = L

If 0 < |x - a| < delta then |f(x) - L| < epsilon

This proof by contradiction is based on Boolean Logic.

Prove that this proof holds also in a multi-valued logical system.

 

[PFanalog57]

This proof by contradiction is based on Boolean Logic.

Prove that this proof holds also in a multi-valued logical system.

You appear to be implying that Boolean logic is context dependent? ...Interesting.

It seems that many valued logic must be formulated in terms of a stable 2-valued logic background.

Suppose the limit as n-->oo, s_n = s, in the classical sense. It must then be demonstrated that s_n - s is infinitesimal for all infinite n. That is to say, for any epsilon > 0 and for any infinite natural number n , it must be proved that |s - s_n| < epsilon.

For any given epsilon > 0 in R there exists a natural number v in N such that

|s_n - s| < epsilon for n > v, n is an element of N.

For all x, if x is an element of N and x > v then |s_x - s| < epsilon.

Since any infinite natural number is greater than v it can be deduced that |s_n - s| < epsilon for all infinite n.


so if I flip a coin, it will be Heads H, or not-Heads, ~H

So if the coin lands on its side, it is still H or ~H, this being the case that it is ~H. Absolutely true.

So if I go for a more specific multivalued logic it becomes H, ~H, S.

H or ~H is still true.

H or ~H or S is just adding more specification...?