Are there any such symbolic representations of Pi or ex?

[RSKueffner]

Howdy folks.

The Gelfond-Schneider Constant 2^21/2 is transcendental. Of my understanding, transcendental numbers are a special case of irrational numbers in the sense of how they may or may not be derrived. This being the case, how is it that many infinite series are stated with such confidence to be transcendental series. For example Sin(x) or Cos(x).

Furthermore, I don't require a lecture on it, but some material for reasearch would be appreciated - how does one prove what an infinite series converges to. I understand this is dependent on the series itself but I'm interested in proving more than the fact that the series converges or diverges.

Lastly, 2^21/2 being transcendental - are there any such symbolic representations of Pi or e^x?

Danke, Kueffner

 

[HallsofIvy]

I am sorry but I really don't understand what you are asking. For one thing, Sin(x) and Cos(x) are not series at all, much less "transcendental series". And as for "symbolic representations of Pi or e^x", Pi is a "symbolic representation" and e^x is a function, not a number.

 

[RSKueffner]

For example:
Sin(X) = [tex]\sum^{\infty}_{N=0}\ (-1)^N \frac{X^{2N+1}}{(2N+1)!}[/tex]
Cos(X) = [tex]\sum^{\infty}_{N=0}\ (-1)^N \frac{X^{2N}}{(2N)!}[/tex]
e^X = [tex]\sum^{\infty}_{N=0} \frac{X^N}{N!}[/tex]

Sine and Cosine may also be expressed in terms of e and so on, so forth.

Pi may be a symbolic representation, but it represents some value. Furthermore, e^X is a function, but said function returns a value per input of X, for example 1.

Symbolic representation wasn't the correct term. Yes, Pi is a symbol. What I mean is, 2^1/2 is a symbolic representation of 1.41421356...

You may even think of that as a function, but is there any ratio of irrational numbers from which you may derive Pi or e that is not an infinite series or limit?

 

[matt grime]

I'm afraid you're still not making much sense to me. I can write pi as the ratio of two irrational numbers, but so what? You need to define what you mean by such terms as 'symbolic representation' and 'derive', though I suspect a lot of it is subjective (pi and e are the integrals of certain functions over certain intervals, for example, would that satisfy you?).I'm still none the wiser as to what you meant by sin(x) being a transcendental series. It is a transcendental function, if that is what you mean. Perhaps you should search (e.g. the wolfram site) for material on algebraic numbers/functions, and transcendental numbers/functions.

 

[CRGreathouse]

You may even think of that as a function, but is there any ratio of irrational numbers from which you may derive Pi or e that is not an infinite series or limit?

What do you mean "is not an infinite series or a limit"? I can certainly find [itex]a,b\in\mathbb{R}\setminus\mathbb{Q}[/itex] with [itex]a/b=\pi[/itex], but what does that condition mean for a and b?

 

[BJRowlett]

I think what RSKueffner is looking for, with regard to Pi, is a symbolic representation for the exact value, which isn't a limit or series.

For example, approximations of Pi:
22/7 ; 333/106 ; 355/113 ; etc...

These symbolic representations are only approximations, unlike 2^sqrt(2), which represents the number in its entirety.

We could write Pi as a series:
4 * Sum(n=0, Infinity) ((-1)^n)/(2n+1)
= 4*( 1 - (1/3) + (1/5) - (1/7) + (1/9) - (1/11) ... )

Or as a limit (which since LaTeX isn't working correctly, I'll provide a link instead):
http://functions.wolfram.com/Constants/Pi/09/0007/


As for Sine and Cosine, both can be expressed as a series:
Sine(x) = Sum(n=0, Infinity) (((-1)^n) * x^(2n+1)) / (2n+1)!
= x - (x^3)/3! + (x^5)/5! - (x^7)/7! ...

Cosine(x) = Sum(n=0, Infinity) (((-1)^n) * x^(2n)) / (2n)!
= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! ...


I personally, do not know the answer, but maybe this clarifies for someone who does..

 

[matt grime]

That is just semantics: 2^sqrt(2) is no more no less of a symbolic representation of a number than \pi is.

 

[RSKueffner]

As is Phi for 1.618... or (1+5^1/2)/2

Rather than Pi being 3.14... I am looking for the equivalent (1+5^1/2)/2 component if such one exists.

 

[mathman]

As is Phi for 1.618... or (1+5^1/2)/2

Rather than Pi being 3.14... I am looking for the equivalent (1+5^1/2)/2 component if such one exists.

Pi is transcendental. There is no algebraic expression for it.

 

[chronon]

What could be asked is whether pi can be the result of an expression involving algebraic numbers and exponentiation. Presumably not. One could then go on to ask whether it is the solution of an equation involving such expressions.

 

[matt grime]

As has been said many times in this thread (including by the OP), pi is transcendental, so no it is not the root of any polynomial with integer coefficients (and this implies any polynomial with algebraic coefficients). Of course the terms 'result' and 'involve' are vague, so you may have something else in mind (if so you should state it clearly).

 

[chronon]

As has been said many times in this thread (including by the OP), pi is transcendental, so no it is not the root of any polynomial with integer coefficients (and this implies any polynomial with algebraic coefficients). Of course the terms 'result' and 'involve' are vague, so you may have something else in mind (if so you should state it clearly).

2^21/2 is clearly an expression involving algebraic numbers and exponentiation and is also transcendental

 

[matt grime]

Erm, what you *really* wrote there is

exp{sqrt(2)*log(2))

of course: that is the only really way to make that symbol 2^sqrt(2) make sense. That or taking limits of rational powers of 2. So once more it is reduced to you needing to define what you mean. If you're allowing that, then I can write pi as 2*arcsin(1).

 

[chronon]

Erm, what you *really* wrote there is

exp{sqrt(2)*log(2))

of course: that is the only really way to make that symbol 2^sqrt(2) make sense. That or taking limits of rational powers of 2. So once more it is reduced to you needing to define what you mean. If you're allowing that, then I can write pi as 2*arcsin(1).

I mean the smallest subset of the reals such that 1 is in the set and if x and y are in the set then so is x#y where # can represent +,-,*,/ and ^. (Maybe it's easier to restrict it to x>0 in the case of x^y). One can then look at the set of solutions to equations which involve a finite combination of integers, variables and +,-,*,/, ^.

 

[RSKueffner]

What could be asked is whether pi can be the result of an expression involving algebraic numbers and exponentiation. Presumably not. One could then go on to ask whether it is the solution of an equation involving such expressions.

I could further comment however I feel this sums it up quite nicely.

Erm, what you *really* wrote there is

exp{sqrt(2)*log(2))

of course: that is the only really way to make that symbol 2^sqrt(2) make sense. That or taking limits of rational powers of 2. So once more it is reduced to you needing to define what you mean. If you're allowing that, then I can write pi as 2*arcsin(1).

One could also write Pi as 4*ArcTan(1) or Ln(-1)/i

I am asking for a non recursive method to define pi in terms of algebraic expression. The sqrt(x) is a none integer coefficient. Furthermore, the method that you bring up is only for integer exponentiation.

For clarification, what I meant about recursive methods is this. You can define the Arc Functions in terms of Pi, or you can define Pi in terms of the Functions. Either way you need a definition of one to define the other. The problem with this is both our definitions, those of Pi, and those of the Arc Functions are infinite series. There exist other methods derived from these or methods that exhibit similar issues. For example an infinite product, etc.

Without expounding further, what I originally intended to ask was summed up exactly by chronon.

If needed, I can provide some examples that get quite complicated when attempting to eliminate infinitely iterating functions.

EDIT: Furthermore, I shall disprove what you (matt grime) wrote by simple example.
exp(sqrt(2)*log(2)) / (2^sqrt(2)) != 1

EDIT: I interpreted log as base 10 and quickly disregarded your statement. I apologize, I was in a rush earlier. Nonetheless, could you then further define your point of making this obvious?

 

[matt grime]

EDIT: Furthermore, I shall disprove what you (matt grime) wrote by simple example.
exp(sqrt(2)*log(2)) / (2^sqrt(2)) != 1

EDIT: I interpreted log as base 10 and quickly disregarded your statement. I apologize, I was in a rush earlier. Nonetheless, could you then further define your point of making this obvious?

I'm confused: you want me to further define what to make what obvious? That log means base e? Well, log always means base e in maths except in the case when you may want to use base 2, but context makes it clear. If you're an engineer you might write ln I suppose, but mathematicians don't, as a rule. Further I thought it was obvious what log meant there since that is just the definition of exponentiation.

 

[RSKueffner]

I asked for you to define your point. Why is this the only way to understand it? It is not my friend. Again, the definitions of these functions are not absolute. You can define it as exp(sqrt(2)*log(2)) and worry about your definitions of e and ln, or you can define it as (2^sqrt(2)) and worry about your definition of a fractional exponent. Either way you calculate it rather interpret it symbolically. exp(sqrt(2)*log(2)) makes no more (nor less) sense than (2^sqrt(2)) as they are the same thing.

The mathematicians comment was a fairly large generalization by the way and is indeed not the case. Perhaps we can get back to the main topic however; after all, all of this has been addressed.

What could be asked is whether pi can be the result of an expression involving algebraic numbers and exponentiation. Presumably not. One could then go on to ask whether it is the solution of an equation involving such expressions.

Perhaps you may verify if this is the case by your experience. Rather, do you know of any valiant attempts if even they were unsuccessful?