# Countable and uncountable sets

#### Loren

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
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 $\frac12$.

#### mobel

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
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.

#### 1ucid

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.

#### abdallahhammam

card(N)=card(Q)=card(set of algebraic)<

card(R)=card(set of transcedental).

#### v8archie

Math Team
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

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.

#### Loren

So the relative density of irrational elements to rational elements is equal to the relative density of transcendental elements to algebraic elements?

#### mobel

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.