Are there more real numbers than natural numbers between 0 and 1?

@KevinWZG said in #1:

Please give proof.
(0,1) for natural numbers, set is empty
(0,1) for real numbers it contrains {2/3,2/4,2/5} that set constains atleast 3 elements
3>0
That ends proof

@tcip said in #9:
Those things are correct, it is litterally given by definition

The smallest positive real number is (that fits on my screen)

0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

@Damkiller25 said in #11:

(0,1) for natural numbers, set is empty
(0,1) for real numbers it contrains {2/3,2/4,2/5} that set constains atleast 3 elements
3>0
That ends proof

going with this that means there are more square numbers than natural numbers. but we can make a 1:1 set up to infinity

1:1 2:4 3:9 4:16 5:25 6:36 7:49 ect.

because we can make a 1:1 ratio they are equal in amount.

@glbert said in #4:

Seems like you (and everyone else in this thread except @JFbin85, frankly) should watch this video, which talk about this exact problem, because your "proof" is unfortunately wrong. https://www.youtube.com/watch?v=_cr46G2K5Fo

@tcip @PenguinWhack271828e @Damkiller25 by obvious common sense, OP misformulated his title question. He means:

"Are there more real numbers between 0 and 1 than natural numbers"?

So just shift "between 0 and 1" a bit front and you have a question worth discussing (although there's still not much to discuss, there's still incredibly way more reals in [0,1] than natural numbers)

sry i meant more real numbers then rational (google translate sucks at this)

Most proofs given by previous posts are wrong, as @aentrenus stated.

No.of rational numbers = No.of naturals
You can correspond every rational number with a natural.

However, there ARE more reals than naturals.

Set of natural numbers is 'countable infinity'.
Set of real numbers is 'uncountable infinity'.

This is because of the diagonal argument, made by Cantor.
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
https://commons.wikimedia.org/wiki/File:Diagonal_argument_01_svg.svg#/media/File:Diagonal_argument_01_svg.svg
https://www.youtube.com/watch?v=OxGsU8oIWjY&t

@Cedur216 said in #16:

@tcip @PenguinWhack271828e @Damkiller25 by obvious common sense, OP misformulated his title question. He means:

"Are there more real numbers between 0 and 1 than natural numbers"?

So just shift "between 0 and 1" a bit front and you have a question worth discussing (although there's still not much to discuss, there's still incredibly way more reals in [0,1] than natural numbers)

So I am not wrong with what I wrote?

@tcip reading your post more carefully, I see you are technically wrong with your line of thought, since as you state yourself, "you're not a math wizard".

Your logic (like any other layman's) is: A is a proper subset of B (that is, set B contains all of set A and even more) --> "there are more numbers in B than in A". By that logic, you would also come to give a response like "there are more rational numbers than natural numbers", which is false.

Infinite cardinalities are hard to grasp. We say that infinitely large sets have the same cardinality (or "size", proverbially), if there is a 1:1 correspondence between them: a mapping that sends all elements of A to all elements of B, without missing any element of B, but without using any element of B more than once at the same time. This is also called a "bijection". And with infinite sets, you can sometimes construct bijections even when one set is a proper subset of another.

Take the subset of even naturals as an example (call it E, while all naturals are called N). I will construct a bijection from N to E simply by mapping each n ∈ N to 2n. This obviously covers all of E, but there are no two distinct n, m being mapped to the same even number (as 2n, 2m are also distinct). In a similar way, the integers Z are just as large as the naturals N.
Even the set Z^2 of all pairs of integers is not larger than N itself. If you visualize N^2 as a grid in a coordinate system, then you can draw a "snake curve" that starts in the origin, goes around the angles and doesn't miss any point of the grid. Each n ∈ N will then be mapped to the n-th point of the curve. Since rational numbers are encoded by numerator and denominator (so they're embedded in Z^2), we also see that Q is no larger than N.

But all of this stops working for real numbers, since heuristically, they contain infinite information without any regularity whatsoever. A formal proof goes by Cantor's second diagonalization argument, as laid out in #6.