Proof -There Are The Same Number of Rational Numbers as Natural Numbers

First we discuss how to compare the cardinalities (sizes) of two infinite sets. Then, we outline Cantors counterintuitive 1874 proof of the fact that the cardinality of (number of elements in) the infinite set of Natural Numbers (1, 2, 3, ) is exactly equal to the cardinality of the apparently much larger infinite set of Rational Numbers (all numbers of the form p/q where p is an integer and q is a natural number). Since the set of Rational numbers include, as a proper subset, the entire set of Natural Numbers in the form 1/1, 2/1, 3/1, , this is a surprising result, and an insight into the nature of infinite sets.

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[Ayasam127]

x = 0.999... (start with x being equal to 0.9 reccuring )

10x = 9.999... ( times both sides by 10 )

10x - x = 9.999... - 0.999... = 9x = 9 ( take 1 lot of x from this value to get 9x is equal to 9 )

x=1 ( divide by 9 to get x equal to one. But x is equal to 0.9 reccuring, thus 0.9 reccuring = 1 )

[MeOnStuff]

@dylansweetensen And lets not even get started on the argument that often flairs up when someone (correctly) asserts that 0.9999... = 1.

[MeOnStuff]

@ForeverWiked That's a problem with the pairing system that *you* have chosen.

If you can find just one way to pair up natural numbers and rational numbers, then you have proven that their sets are of equal cardinality.

[ForeverWiked]

If you line up this way though

1/1 matches 2, and 2/1 matches 2, it leaves no match for 1/2 or 3/2

[dylansweetensen]

ITT: People who don't really know set theory trying to prove hundreds of years of philosophical and mathematical thinking wrong.

[MrCheesymonky]

@JamesTR4 first of all, regurgitating everything that you read from books does not show your intelligence >.>.

Secondly, to reply to your question, i believe that your question is wrong. real numbers include irrational numbers, so i dont even need to explain the rest...

[JamesTR4]

@MrCheesymonky First of all, you are not against this man's teaching. This result is a well-established fact of mathematics. You haven't completely understood how a bijective function works to compare two sets of infinite numbers. Sets with infinite numbers are counter-intuitive mainly because you can get the same cardinality even if one set is a proper subset of the other. Did you know that the numbers from 0 to 1 (real) have the same cardinality with the numbers 0 to 100(also real)?

[sponsoredwalk1]

@MrCheesymonky

I recommend reading the start of Kolmogorov's

Introductory Real Analysis for a very short, clear

and understandable proof of this concept, just

make sure you read the definitions properly.

[cventer73]

This man does not understand the concept of an algebraic proof. Also, just because two rational numbers 1/1 and 2/2 are proportional does not mean they are the same rational number and cannot be "skipped" using his vague path algorithm. Even geometric proofs using a triangle need be more rigorous than this. You cannot count infinite series by definition. We don't know how many of either set rational or natural their are. He also talks unbearably slowly, I'm glad he's not my math teacher.

[MrCheesymonky]

@KohrAh2718: that does not reply to my comment >.>...im not saying anything about irrationals...im talking about rational and natural numbers...using your explanation, there are an infinitely more rationals than natural numbers, because the next rational number is unknown. for example, after 1, name the next rational number; you cannot. even between every rational number, there are infinite rationals