Countable and uncountable sets

May 2015
516
32
Arlington, VA
Does the number of all irrational elements divided by the number of all rational elements equal the number of all transcendental elements divided by the number of all algebraic elements?

Are these ratios otherwise comparable?
 

v8archie

Math Team
Dec 2013
7,713
2,682
Colombia
I think you should look at the concept of density. I think the density of each of these sets in the reals is 1. The density of the even numbers in the naturals is $\frac12$.
 
Dec 2013
1,117
41
I think you should look at the concept of density. I think the density of each of these sets in the reals is 1. The density of the even numbers in the naturals is $\frac12$.
Shut up! You know nothing about topology field.
So shut up! shut up! shut up!
 

v8archie

Math Team
Dec 2013
7,713
2,682
Colombia
If you feel that there is an error in what I have suggested, you should correct it or explain why it is an error.

Otherwise, you try to act your age and avoid hurling childish abuse. In fact, if you have nothing of substance to add to the thread you really ought to consider taking your own advice.
 
Aug 2015
37
4
Montenegro (Podgorica)
Shut up! You know nothing about topology field.
So shut up! shut up! shut up!
don't be aggressive, he is just telling his opinion.
 
Aug 2016
273
32
morocco
card(N)=card(Q)=card(set of algebraic)<

card(R)=card(set of transcedental).
 

v8archie

Math Team
Dec 2013
7,713
2,682
Colombia
Actually, on reflection, the density of the countable sets in the reals is perhaps zero. But that's why I suggest looking it up.

The point is that division isn't defined for the transfinite cardinalities as far as I am aware.
 

SDK

Sep 2016
805
545
USA
Does the number of all irrational elements divided by the number of all rational elements equal the number of all transcendental elements divided by the number of all algebraic elements?

Are these ratios otherwise comparable?
This is a great question that doesn't really have an easy answer. Many of the intuitive aspects of numbers break down at infinite sets. For example, we use the word cardinality to discuss the size of these sets and the cardinality of the algebraic numbers is equal to that of the rationals. However, even though they are all infinite sets, the cardinality of the irrationals is greater than that of the rationals/algebraics and equal to that of the transcendentals.

There is a notion of multiplying and dividing infinite cardinal numbers but I don't know much about it and that doesn't seem like what you are interested in. I think v8archie is probably correct and you are talking about density. If this is the case you will probably have to ask your question more precisely.
 
May 2015
516
32
Arlington, VA
So the relative density of irrational elements to rational elements is equal to the relative density of transcendental elements to algebraic elements?
 
Dec 2013
1,117
41
If you feel that there is an error in what I have suggested, you should correct it or explain why it is an error.

Otherwise, you try to act your age and avoid hurling childish abuse. In fact, if you have nothing of substance to add to the thread you really ought to consider taking your own advice.
Why do not say I do not know?
Oh! An auto-elected teacher who does know? what a shame!
Shut up if you do not idea about what density is. Shut up! Shut off your mouth!
I will not correct you even if you are obviously wrong ( I mean anyone can see that you are wrong). It is up to the moderators to tell you that you are wrong. That is the only way to give credibility to the forum. It is not the first time that you come big absurdities. If the moderators were aware of what you are doing they have to banish you for ever. You are more damaging to the forum than a crank or a troll.