## Line of real numbers-transcendental numbers

Hi everybody,
I would like to ask two things:

1)What is the line of real numbers? Is it just a graphical way of representing the set of real numbers?

2)How is the existence of transcendental numbers explained? I mean, if we look at the number line, and thinking of the Dedekind cuts, why only very few of the irrationals are transcendental numbers? Do we know how many transcendental numbers exist? And is there some explanation for their position in the number line?

Thanks
 Recognitions: Gold Member Science Advisor Staff Emeritus 1) Yes, the "number line" is just a graphical way of representing the set or real numbers. 2) "why only very few of the irrationals are transcendental numbers?" Excuse me? In a very specific sense, almost all real numbers ARE transcendental! That is, the set of all transcendental numbers is far larger than the set of algebraic numbers (non-transcendental numbers which includes all rational numbers). As far as "how many irrational numbers exist": an "uncountable" number. Basically, "countable" means that there is a one-to-one correspondence between the set and the set of all counting numbers (1, 2, 3, etc.). "Uncountable" means there are far too many to "count"- to set into a correspondence with even all counting numbers. Their position? In any interval of real numbers, no matter how small, there exist and infinite (indeed "uncountable") number of irrationals and and infinite (but countable) number of rationals.
 Recognitions: Gold Member Science Advisor 1) It's a geometrical way of represnting the real numbers so we have a line where each poitn correponds to a real number. Of course when preople talk about the rela line quite often they're not specifically refering to the geomtrical represemtation, instead they are talking about the real numbers as an ordered set/metric space/topological space/etc. 2) As Hall ofIvy says, the transcendental numbers have the cardinality of the continuum (i.e. the same cardianlity as the rela numbers) wheras the algebraic (non-transcendental) numbers have a cardinailty of aleph-0, the same cardianltiy as the natural numbers.

## Line of real numbers-transcendental numbers

I made a mistake in my initial post. I have edited it, so the question is how many transcendental numbers (and not just irrational) exist? However, you mention that almost all real numbers are transcendental! I don't think that is correct. Transcendental numbers examples: pi, e ...
And I think that irrationals, like rationals are a countable set, but i am not sure of that.
Actually i have just found this from MathWorld:
From Gelfond's theorem, a number of the form a^b is transcendental (and therefore irrational) if a is algebraic<>0,1 and b is irrational and algebraic.

 Quote by C0nfused I made a mistake in my initial post. I have edited it, so the question is how many transcendental numbers (and not just irrational) exist? However, you mention that almost all real numbers are transcendental! I don't think that is correct. Transcendental numbers examples: pi, e ... And I think that irrationals, like rationals are a countable set, but i am not sure of that. Actually i have just found this from MathWorld: From Gelfond's theorem, a number of the form is transcendental (and therefore irrational) if a is algebraic , 1 and b is irrational and algebraic.
Show that the set of all algebraic real numbers is countable (easy). A transcendental number is a non-algebraic real number. Thus, imply that the set of transcendental numbers is uncountable.
Similarly, show that the set of all rational numbers is countable. Thus imply that the set of all irrational numbers, as the complement of the rationals in the reals, is uncountable.

 Quote by hypermorphism Show that the set of all algebraic real numbers is countable (easy). A transcendental number is a non-algebraic real number. Thus, imply that the set of transcendental numbers is uncountable. Similarly, show that the set of all rational numbers is countable. Thus imply that the set of all irrational numbers, as the complement of the rationals in the reals, is uncountable.
You are right, rationals are countable and reals are not, so irrationals can't be countable. As for the set of all algebraic real numbers, if it is countable, then transcendentals are not countable.
 Then to show that the almost all real numbers are irrational, show the rationals have measure zero. Start with rationals $$h/k$$ with $$11$$ which can be enclosed in an interval in the same way as above. Extension to zero, 1, and the negative rationals is obvious.