Irrational numbers vs. Transcendental numbers

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Discussion Overview

The discussion centers on the distinctions and relationships between irrational numbers and transcendental numbers, exploring definitions, examples, and implications within mathematical theory. Participants examine the characteristics of these types of numbers and question the boundaries of their definitions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose that all irrational numbers must be transcendental, based on the idea that transcendental numbers are non-repeating and infinite.
  • Others clarify that irrational numbers, such as √2, are not transcendental because they can be solutions to polynomial equations, while transcendental numbers, like π, cannot.
  • There is a discussion about the definitions of irrational and transcendental numbers, with some participants expressing confusion over the implications of these definitions.
  • One participant questions whether the lack of repetition in irrational numbers is a defining characteristic.
  • Another participant raises a question about the relationship between continued fractions for irrational and transcendental numbers, as well as the construction of transcendental functions from polynomial functions.
  • Participants discuss the nature of polynomial equations and their coefficients, emphasizing that transcendental numbers cannot be derived from polynomials with rational or algebraic coefficients.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of irrational and transcendental numbers. There is no consensus on whether all irrational numbers can be classified as transcendental, and the discussion remains unresolved regarding the nuances of these definitions.

Contextual Notes

Some definitions and assumptions about the nature of irrational and transcendental numbers are not fully explored, leading to potential misunderstandings. The discussion also touches on the limitations of polynomial equations in relation to transcendental numbers.

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It would seem that an irrational number would have to be a transcendental number. If a transcendental number is a number which goes on infinitely and never repeats, then all irrational numbers would have to be transcendental, because if they repeated then you could find a fraction doing the whole x = .abcdef... 1000000x=abcdef.abcdef... and so forth.
Are there any counter-examples?
 
Last edited:
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First off:
Irrational means 'cannot be expressed as a fraction'. For example [tex]\sqrt{2}[/tex] is an irrational number.

Transcendental means is not a solution to any equation that contains only rational numbers. For example, [tex]\pi[/tex] is a transcendental number. [tex]\sqrt{2}[/tex] is not a transcendental number.
 
Originally posted by NateTG
First off:
Irrational means 'cannot be expressed as a fraction'. For example [tex]\sqrt{2}[/tex] is an irrational number.

Transcendental means is not a solution to any equation that contains only rational numbers. For example, [tex]\pi[/tex] is a transcendental number. [tex]\sqrt{2}[/tex] is not a transcendental number.
Ah. Then I believe my problem was in my defintions, however, I was under the impression that transcendental meant it never repeated or whatnot. I guess this is more of a symptom. so, is root 2 not transcendental because it solves x^2 = 2? Wouldn't that make pi merely irrational, as it solves various series that converge at it, like x = 1 + 1/3 - 1/5 or however it goes.
Thanks for your time!
 
Sorry, let me me more clear.
Transcendental numbers can never be expressed as the roots of polynomial equations. (Equations involving addition, multiplication, subtraction or division. No infinite series, logarithms, or trig functions.)
 
Originally posted by NateTG
Sorry, let me me more clear.
Transcendental numbers can never be expressed as the roots of polynomial equations. (Equations involving addition, multiplication, subtraction or division. No infinite series, logarithms, or trig functions.)
Ah! That clears that up.
So is the lack of any repeating trademark to irrational numbers as a whole?
 
Well, "lack of any repeating" isn't a good description, but that's essentially it, yes.
 
Isn't that also (indirectly) stating that trancendental numbers appear in their own definition?
 
Originally posted by On Radioactive Waves
Isn't that also (indirectly) stating that trancendental numbers appear in their own definition?

I don't understand what you mean.
 
Originally posted by NateTG
Sorry, let me me more clear.
Transcendental numbers can never be expressed as the roots of polynomial equations. (Equations involving addition, multiplication, subtraction or division. No infinite series, logarithms, or trig functions.)

It's important to add that these polynomials have integer, or rational, or algebraic coefficients. In other words you can't get transcendental numbers from polynomials with coefficients that aren't transcendental.
 
  • #10
What is the relationship between continued fractions for irrational numbers and how exactly does this differ from the continued fractions of transcendental numbers?

Also wondering: transcendental functions (trig and log functions, infinite series): completely impossible to construct out of polynomial functions with 100% accuracy?
 
  • #11
selfAdjoint said:
It's important to add that these polynomials have integer, or rational, or algebraic coefficients. In other words you can't get transcendental numbers from polynomials with coefficients that aren't transcendental.

Yes, otherwise x = pi would satisfy criteria. Also in same spirit x = sqrt(2) square both sides and you recover 2. x = pi what do you do now? what do you recover using allowed operations on polynomials? what is inside pi and how do you get it out? LOL

1/sqrt(2) rationalize the denominator...very easy.

1/pi rationalize the denominator...you become FAMOUS!
 

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