Are there any such symbolic representations of Pi or ex?

In summary, the Gelfond-Schneider Constant 221/2 is a transcendental number. Transcendental numbers are a special type of irrational numbers that cannot be derived from any algebraic expression. This raises the question of how infinite series, such as Sin(x) and Cos(x), can confidently be stated as transcendental series. To further understand this concept, research is needed on proving the convergence of infinite series. Lastly, it is worth noting that while Pi and e may have symbolic representations such as 2^sqrt(2) and (1+51/2)/2, they are still transcendental numbers and cannot be derived from any algebraic expressions.
  • #1
Howdy folks.

The Gelfond-Schneider Constant 221/2 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, 221/2 being transcendental - are there any such symbolic representations of Pi or ex?

Danke, Kueffner
 
Physics news on Phys.org
  • #2


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 ex", Pi is a "symbolic representation" and ex is a function, not a number.
 
  • #3


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]
eX = [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, eX 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, 21/2 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?
 
Last edited:
  • #4


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.
 
  • #5


RSKueffner said:
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?
 
  • #6


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..
 
  • #7


That is just semantics: 2^sqrt(2) is no more no less of a symbolic representation of a number than \pi is.
 
  • #8


As is Phi for 1.618... or (1+51/2)/2

Rather than Pi being 3.14... I am looking for the equivalent (1+51/2)/2 component if such one exists.
 
  • #9


RSKueffner said:
As is Phi for 1.618... or (1+51/2)/2

Rather than Pi being 3.14... I am looking for the equivalent (1+51/2)/2 component if such one exists.
Pi is transcendental. There is no algebraic expression for it.
 
  • #10


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.
 
  • #11


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).
 
  • #12


matt grime said:
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).
221/2 is clearly an expression involving algebraic numbers and exponentiation and is also transcendental
 
  • #13


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).
 
  • #14


matt grime said:
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 +,-,*,/, ^.
 
  • #15


chronon said:
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.

matt grime said:
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?
 
Last edited:
  • #16


RSKueffner said:
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.
 
  • #17


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.

chronon said:
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?
 

1. What is the significance of Pi and ex in mathematics?

Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning it cannot be expressed as a simple fraction. Ex (e) is another mathematical constant that represents the base of the natural logarithm. It is also an irrational number and has many applications in calculus and other areas of mathematics.

2. Are there any real-life applications of Pi and ex?

Yes, both Pi and ex have numerous real-life applications. Pi is used in geometry, physics, and engineering to calculate the circumference, area, and volume of circles and spheres. Ex is used in finance, biology, and physics to model exponential growth and decay, as well as in probability and statistics to calculate compound interest and continuous probability distributions.

3. Can Pi and ex be represented symbolically?

Yes, Pi and ex can be represented symbolically using various mathematical notations and symbols. For example, Pi is often represented as the Greek letter π, while ex can be written as e. Additionally, both constants can be expressed using infinite series or continued fractions.

4. Is there a relationship between Pi and ex?

Yes, there is a mathematical relationship between Pi and ex known as Euler's identity. It states that e^(iπ) + 1 = 0, where i is the imaginary unit. This relationship connects the two constants and is considered one of the most beautiful and elegant equations in mathematics.

5. Are there any attempts to calculate the exact values of Pi and ex?

Yes, mathematicians and computer scientists have made numerous attempts to calculate the exact values of Pi and ex. However, due to their irrationality, the digits of both constants continue infinitely without repeating. As of now, Pi has been calculated to over 31 trillion digits, while ex has been calculated to over 8 trillion digits.

Suggested for: Are there any such symbolic representations of Pi or ex?

Replies
1
Views
788
Replies
16
Views
2K
Replies
2
Views
849
Replies
3
Views
2K
Replies
33
Views
3K
Replies
1
Views
1K
Replies
1
Views
1K
Back
Top