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

[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 )

[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

[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.

[dylansweetensen]

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

[MeOnStuff]

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

[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...

[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

[MrCheesymonky]

although i like this man's teaching, i cant help but be against it.

The "proof" he was talking about for 8 minutes did anything but prove that there are the same amount of rational and natural numbers.There are clearly more rational numbers, because he even said that rational numbers INCLUDE ALL natural numbers...therefore, anything added to it means there are more..

plus. there are an infinite amount of rational numbers between each natural number...do i have to say much more?>.>...

[KohrAh2718]

@MrCheesymonky

You can count 1,2,3... with natural numbers just like you can count 2,4,6... with the even naturals.

Infinite is infinite is infinite, if you can count it, and know what the 'next' element is.

Defining rationals as p and q, you can count p and q forever.

Transcendental irrationals are far more infinite than the naturals, rationals, or algebraic numbers, because nobody knows what the next transcendental number is.

[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.

[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)?

[gmussina]

you actually need two requirements for two sets having the same cardinality. the first one is one-to-one, but you actually need onto-ness of the second set(unless you are doing wlog)

[fahgsdhjh]

He said "one to one correspondence" not just "one to one" which means one to one and onto (bijection)

One to one correspondence means the relation is one to one both ways

[agentredlum]

Professor, you say that using this 'diagonal' you get numbers that possibly repeat. I would like to point out that EVERY number in that box repeats an infinite amount of times! (except zero) You have to hop and skip over an infinite number of elements to make the theorem work not to mention reducing an infinite amount of vulgar fractions in order to see if they are already on the list! Every time you make a move on your 'diagonal' you have to check the rational to make sure its not a repeat.

[SigmaScales90]

Hmm, I'm still confused. At the end of the proof, you "match" 1/3 from Q with 7 from N. To my untrained mind, it would seem that you "run out" of Natural numbers at a faster rate than the Rational numbers. Is that truly the end of the proof?!

[M1cLovin]

There are infinite number of natural numbers, so that means we never run out of them, so we will keep matching them infinitely