Did you not read royalchicken's post? This question has been asked and answered, and I outline my solution in my reply. It probably has been published in a textbook, but who cares? What hasn't? Feel like solving the Navier-Stokes equations, Mr. Original?

But I did say I would provide my solution if asked, and as I am not a weasel, here it is (in 2 parts):

(A) natural numbers/rational numbers

List the non-negative rational numbers in a matrix as follows:

0/1 0/2 0/3 ...

1/1 2/1 3/1 ...

2/1 2/2 2/3 ...

3/1 3/2 3/3 ...

...

This matrix contains all quotients (p/q) of the natural numbers, and hence contains all the positive rational numbers (some more than once). Now we will order this matrix using the following algorithm:

1. Assign an index to each matrix entry in the form (a,b), where "a" is the row number and "b" is the column number.

2. Start with entry (1,1), which in the above matrix is 0/1 = 0.

3. For the next ordered rational number, make the previous number negative. Our first number, 0, is a special case where -0 = 0, so skip it (lest we double count).

4. To move to the next matrix entry, subtract 1 from both "a" and "b" so you move in a diagonal direction. If (b-1) = 0, move to (1,b), where "b" is the lowest column number not covered yet. In this case, 1-1=0, so we move to (1,2).

5. If the new entry is equal in magnitude to an entry already covered, remove that entry. 0/2 = 0 which was covered in step 2, so remove that entry (again, lest we double count).

6. Repeat steps 3 through 6, creating an ordred list of all the rational numbers where each number is listed only once. By ordering them this way, we create a list that extends inifitely in one direction from a matrix that extends infinitely in two directions.

7. Number the first entry ("0"ðŸ˜‰ in this list with a the natural number "1". Now number each successive ordered rational number with the next natural number, going in order of increasing magnitude. This pairing (mapping) of natural numbers and rational numbers is 1-1 and onto, meaning that each entry in both lists corresponds with 1 and only 1 entry in opposite list. Therefore, the "number" of elements, or more properly the cardinality of each list, is the same.

Therefore, the number of natural number and rational numbers is the same.

(B) real numbers/rational numbers:

We begin by creating two columns (1 and 2). In column 1, we list all the natural numbers in order of increasing magnitude. In column 2, we list the decimal expansions of the real numbers between 0 and 1 with no repeats (order does not matter). All decimals will be of infinite length, for example 0.5 will be written as 0.5000000... with an infinite number of 0's after the 5, and repeating decimals like 1/3 = 0.33333... will repeat infinitely.

It would appear that both lists have the same number of elements, and hence are of the same size, but this is not the case. It is possible to write a number in column 2 that has no corresponding number in column 1, and we do this as follows:

1. The new number (which lies between 0 and 1) has as the number in it's first decimal place the number in the first decimal place of the first number in column 2.

2. The number in it's second decimal place is equal to that in the second decimal place of the second number, the number in it's third decimal place is equal to that in the third decimal place of the third number, etc...

To make this slightly more concrete, let's use an example list:

1 2

1 0.7148302865...

2 0.3333333333...

3 0.5090909090...

4 0.8291287641...

... ...

Moving diagonally down and right from the first decimal place of the first number ("7"ðŸ˜‰, our new number would start 0.7391...

3. Now we change the new number in each decimal place as follows: add 1 to the number, unless the number is 9 in which case 9 becomes 0.

Our above number 0.7391... would become 0.8402...

This new number is a real number between 0 and 1 different from every other number in column 2 in at least one decimal place, therefore there is not a 1-1 and onto mapping from the natural numbers to the real numbers between 0 and 1. It is also clear that the list of real numbers between 0 and 1 has at least one more element. Therefore, there are more real numbers between 0 and 1 than rational numbers (which follows from the equal cardinality between natural and rational numbers established in (A) above), and therefore there are more real numbers in total than rational numbers.