Transcendental vs. SUPER transcendental?

  • Thread starter 1MileCrash
  • Start date
In summary: Which is to say, in the same sense that e, pi and the golden ratio are.In summary, there are different classes of transcendental numbers - algebraic and transcendental. Algebraic transcendental numbers can be expressed as the solution to a polynomial equation with integral coefficients, while transcendental numbers cannot. However, the exact number of transcendental numbers is uncountably infinite and cannot be fully enumerated.
  • #1
1MileCrash
1,342
41
Is there any differentiation between transcendentals that can be expressed algebraically (through finite operations) and those that can't?

IE, the golden ratio is transcendental but is also [1 + sqrt(5)]/2

pi or e cannot be defined this way.

So are there different classes of transcendentals, or do we view these types of numbers equally?
 
Mathematics news on Phys.org
  • #2
The golden ratio is not transcendental. It is the solution to x2-x-1=0.
 
  • #3
Yes, I knew that. Then it is clear that my understanding of what constitutes a transcendental number is wrong.

I always thought it was an irrational number that could not be brought to a rational through an operation on the number, but I can see that's incorrect. (what I mean is that, sqrt(2) is irrational, but squaring that gives me 2, therefore sqrt(2) is not transcendental.)

Thanks!
 
  • #4
1MileCrash said:
Is there any differentiation between transcendentals that can be expressed algebraically (through finite operations) and those that can't?

IE, the golden ratio is transcendental but is also [1 + sqrt(5)]/2

pi or e cannot be defined this way.

So are there different classes of transcendentals, or do we view these types of numbers equally?

Clearly the g.r. is algebraic, being a root of x^2 - x - 1.

I agree that the g.r. has the property that most people think it's transcendental. In that respect it's a bit like a Grothendieck prime.

One striking characteristic of Grothendieck’s
mode of thinking is that it seemed to rely so little
on examples. This can be seen in the legend of the
so-called “Grothendieck prime”. In a mathematical
conversation, someone suggested to Grothendieck
that they should consider a particular prime number. “You mean an actual number?” Grothendieck
asked. The other person replied, yes, an actual
prime number. Grothendieck suggested, “All right,
take 57.”


http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf
 
  • #5
1MileCrash said:
Is there any differentiation between transcendentals that can be expressed algebraically (through finite operations) and those that can't?

IE, the golden ratio is transcendental but is also [1 + sqrt(5)]/2

pi or e cannot be defined this way.

So are there different classes of transcendentals, or do we view these types of numbers equally?

The golden ratio is an irrational number, as are pi and e.

Irrational numbers can be subdivided into algebraic irrational numbers and transcendental numbers. The former (algebraic irrational numbers) can be expressed as the solution of a polynomial equation with integral coefficients, e.g. like (x^2 - 2 = 0 for √2) and (x^2 - x - 1 = 0 for the golden ratio). The latter (transcendental numbers) cannot be so expressed, but can be expressed as infinite sums or products of algebraic expressions with integer coefficients, or as limits where infinity is involved in some way. Neither sort of irrational number (algebraic or transcendental) has any periodicity in its decimal expansion (or digit expansion in any integer base), so a decimal representation of the number can never be written out in full.

You've made a misstep in calling the golden ratio transcendental. Nevertheless, your question, slightly amended, has value. How many transcendental numbers can we actually express? I'm using express in the sense that we can write down a mathematically true expression with a finite span that equals the number. There are only three ways this can be done:

1) To find a finite expression for the number in terms of a formula (involving infinite sums or limits) using elementary operations such as addition, subtraction, multiplication, division and exponentiation, with integer coefficients. The "famous" ones like [itex]\pi[/itex] and [itex]e[/itex] are covered here.

2) To "construct" the decimal expansion of a number with a predictable but non-periodic pattern. Liouville's constant and the Champernowne constant fall under this category.

3) To "construct" a transcendental number by algebraically transforming a known one. Note that all transcendental numbers formed by such an operation would be algebraically dependent with the original transcendental. As an example, we can form the number [itex]\pi + 1[/itex], which is transcendental, distinct from pi, but still algebraically related to pi. In fact, we can form a whole infinite class of such numbers defined by [itex]\pi + 10^{-n}, n \in \mathbb{Z}^+[/itex], which would each differ in a finite number of decimal places from pi.

Using any of those methods, can we express or even enumerate all the transcendentals? The answer is an emphatic NO. You see, Georg Cantor did some great work in this area. The number of transcendental numbers is uncountably infinite, which means it is impossible to list them out. Even the last set of numbers I constructed using pi has only a countably infinite number of elements. This still falls far shorts of the uncountably infinite number of transcendentals that exist.

The punchline is that there's no need for "SUPER" transcendentals - transcendentals alone are already "super" enough. One can never get a full grasp of how many there really are. Even if one can conceive of all the known transcendental numbers and think of all the algebraic manipulations that one can apply to them, one would still only have enumerated an infinitesimal (using it loosely here) fraction of the transcendental numbers that exist. There will always be an uncountable number of "unsung heroes" among the transcendental numbers. :biggrin:
 
  • #6
To further expound upon the distinction which I think may have motivated your post. There exist solutions to polynomial equations over the integers which cannot be expressed as some finite combination (sum/product/difference/quotient) of integers, plus the extraction of roots. These algebraic irrational numbers are those associated with polynomials which do not have solvable galois groups.

Concisely put, there are roots of polynomials which are only explicitly expressible as some infinite series or continued fraction.
 
Last edited:
  • #7
What about a number such as 2^(sqrt(2))?
 
  • #8
Well yes that is transcendental, but I wouldn't consider it in the class of numbers capable of being written as some finite combination (sum/product/difference/quotient) of integers, plus the extraction of roots.
 

Related to Transcendental vs. SUPER transcendental?

What is the difference between transcendental and SUPER transcendental numbers?

Transcendental numbers are real numbers that cannot be expressed as a solution to a polynomial equation with rational coefficients. SUPER transcendental numbers are a subset of transcendental numbers that have even stronger properties, such as being non-computable or non-constructible.

How can one identify a SUPER transcendental number?

There is no general method for identifying SUPER transcendental numbers. However, some examples of known SUPER transcendental numbers include the Chaitin's constant, the Liouville constant, and the Champernowne constant.

Are there more transcendental numbers or SUPER transcendental numbers?

There are infinitely many transcendental numbers, but only a few known SUPER transcendental numbers. It is believed that there are more transcendental numbers than SUPER transcendental numbers.

What are some applications of transcendental and SUPER transcendental numbers?

Transcendental numbers have applications in various fields, such as computer science, physics, and engineering. SUPER transcendental numbers have been used in the study of randomness and complexity, as well as in cryptography and information theory.

Is there ongoing research on transcendental and SUPER transcendental numbers?

Yes, there is ongoing research in this area, particularly in the study of SUPER transcendental numbers and their properties. Researchers are also exploring the connections between transcendental and SUPER transcendental numbers and other areas of mathematics, such as chaos theory and fractal geometry.

Similar threads

Replies
1
Views
1K
Replies
8
Views
1K
Replies
7
Views
4K
  • General Math
Replies
13
Views
2K
  • General Math
Replies
21
Views
2K
Replies
7
Views
899
Replies
3
Views
5K
  • General Math
Replies
22
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
1K
  • General Math
Replies
5
Views
2K
Back
Top