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?
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.)
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.
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?
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!