How can there be more rational numbers than natural numbers?

[cragar]

this may be a dumb question but I haft to ask it , so if there are an infinite number of natural numbers 1 , 2 ,3 ... , and there are an infinite number of rational numbers in between 1 and 2 how can they both be infinite if I have more rational numbers than natural numbers . can we quantify infinities

 

[dacruick]

infinity with respect to what you are talking about means never ending. The amount of numbers between 1 and 2 is never ending, but there are also a never ending amount of numbers to begin with. So you are wrong to say that you have more rational numbers than natural numbers, because they can both go on forever.

Furthermore, infinity only represents this "never ending" concept. So no, it can not be quantified.

I hope I answered your question.

 

[cragar]

ok i see what ur saying , but if we take all the numbers into account from 0 to infinty iff we add them up let's say were at the nth therm we will have more rational numbers than natural numbers .

 

[HallsofIvy]

There are many different ways to quantify "infinite" depending on what you want to do. One is the distinction between "countable" and "uncountable" infinities. The integers and the rational numbers are both "countable" while the real numbers are "uncountable". In fact the set or all rational numbers is countable while the set of real numbers between, say 0 and .000000001 is uncountable and so, in this sense far larger than the set of all rational numbers.

 

[DaveC426913]

One is the distinction between "countable" and "uncountable" infinities. The integers and the rational numbers are both "countable" while the real numbers are "uncountable". In fact the set or all rational numbers is countable while the set of real numbers between, say 0 and .000000001 is uncountable and so, in this sense far larger than the set of all rational numbers.

Is that what countable means? I guess I've never really gotten it.

So, countable literally means you can say "1, 2, 3, 4, etc...". While you may never complete the set (because it's inifinite), you are counting off the numbers.

Contrarily, you cannot count off real numbers (1, 1.1, 1.2, -oops- 1.11, -oops- 1.00001, etc.)

 

[dacruick]

ok i see what ur saying , but if we take all the numbers into account from 0 to infinty iff we add them up let's say were at the nth therm we will have more rational numbers than natural numbers .

I believe the issue with your way of thinking is that you you are saying that you take all the number into account from 0 - infinity. But you can never reach infinity.

It is my understanding that you are saying since there are an infinite amount of numbers between 0 and 1, and there are an infinite amount of whole numbers, there must be more rational numbers because it is basically infinity multiplied by infinity. But that is trivial. You are treating infinity like a number, but it has to be treated as a concept instead.

If you still aren't understanding I might be able to put it into your terms even though I do not believe this is the right way to be thinking about infinity.

Infinity multiplied by infinity is still infinity. Therefore, there can't be more rational numbers than natural numbers because they both are infinite.

 

[cragar]

ok i got it , i was just trying to think about it from a different point of view.

 

[joelr315]

this may be a dumb question but I haft to ask it , so if there are an infinite number of natural numbers 1 , 2 ,3 ... , and there are an infinite number of rational numbers in between 1 and 2 how can they both be infinite if I have more rational numbers than natural numbers . can we quantify infinities

There are different levels of infinity in Cantors transfinite numbers theory.

The first infinity - "w" (Omega) or Aleph_Null includes all the natural numbers.

The set of all irrational numbers is Aleph_1.

The infinities are set up on a basis of mapping on top of each other. Aleph2 cannot be mapped on a one to one basis onto Aleph1.

For example in Hilbert's Hotel you have an infinity of rooms with an infinity of guests staying.
If a new guest comes everyone moves over by 1 room because w + 1 = w.

If an infinity of new guests shows up everyone moves over to the infinity of even numbered rooms and the new guests stay at the odd rooms.

So w + w = w

I think Alef_2 has been defined as the number of points on any of the possible curved surfaces, including all the unusual surfaces that are possible.

I think this is also = to w tetrated to w. Tetration is the next step after - add, multiply, exponentiate, tetration, pentration, really big ...

Alef_3 is unknown. Maybe some type of 4D or higher dimension object would qualify?

 

[cragar]

interesting , thanks for the response .

 

[LouieHussey]

Cragar, there is a fascinating popular mathematics book on the subject of infinity and transfinite numbers that you might be interesting in looking at, if it is still in print; it is very simple and easy to follow:

Infinity: Beyond the Beyond the Beyond, by Lillian R. Lieber, published by Holt, Rinehart and Winston, 1953.

Lillian Lieber, used to be professor and head of the department of Mathematics at Long Island University.

Louie

 

[cragar]

interesting i will look into it .