Which set is bigger: the integers or the rational numbers?

[d.]

your energy is brilliant. first time i've subscribed to anything in ages.

[SaloCh]

Great video! My only problem would be that you kind of sweeped negative numbers under the rug, but I guess it's not hard to expand this for negative numbers. You only really have to change the sign of the numerator, and start the diagonal on the left. You also avoid the denominator = 0 problem.

[lifeofphyraprun]

Excellent video Mr.Woo!I always hated the idea of Infinity and thought that it was a wrong idea, because it has many contradictions, but now I'm beginning to understand that it is just that we have to get used to the idea and understand the nature of 'endless', and what it means for infinities to be of different kinds and sizes and that the general notions and literal meanings are the enemies of this idea.And then how intuitive some of the things start to seem when the mind is opened up to new facets of thinking and reasoning and leaves its rigid, conservative space and truly develops.I hope, with time I will understand even more.You are great Mr.Woo!My respect!

[combinedmathsbysachithband]

i am lucky to have such a brilliant teacher in the earth..thanks

[umaisshahid]

Sir first of all my English is very weak,
Thank you, sir for providing this world huge beneficial knowledge.
Dear sir, I wanna say you that we have not basic ideas of sets (like integers, real, rational and irrational etc) please guide us for the basic tool of mathematics like sets subsets etc.
please please please please

[manarsalem]

Omg I was on the verge of crying trying to understand this until I'd finally opened your video. Thanks a lot for uploading!

[grahamzibar]

Why is there a whiteboard atop another whiteboard?

[richardsaid]

Mr.woo I can agree with the diagonal method of matching to an extent and although, I typically do agree that the conclusions you come to are usually correct and likely well documented studies within professional mathematical literature, I can not agree to the matching of the natural numbers with the even number example that you showed. This is due to the fact that you matched up all the numbers in an orderly and countable way (1-2, 2-4, 3-6, 4-8....) I believe this proof of natural set of numbers = even set of numbers (or any other set for that matter) only works for depending on where you BEGIN to match the set's and ongoing. Sure you may BEGIN to match all the numbers as it's given in order however if you began to match like

1 2 (connect 2 with 2)
2 4 (connect 4 with 4)
3 6 (connect 6 with 6)
4 8 etc
5 10 etc
. . etc
. . .
. . .

connecting them diagonally will still be able to successfully match each even number with one of the natural numbers and STILL have unmatched natural numbers left. I am aware that a counter argument could be "well this is not real matching since the lines within the set must be matched in order" or "that does not undermine the fact that starting at the beginning will still allow you to match up each value". However those justifications for objecting what I have presented just seams to be illusionary and a bit dishonest form of reasoning as to why one argument should still have credence over another. There simply needs to be more careful consideration and logical consistency as to why this proof actually holds rather than some other proof that can utilize a similar method (different in quality of order) and yet breed different results (I apologize if I have straw manned or not brought up an actual counter argument one may have, these were simply the first that came to mind).

[laurak]

If there are repeats in the rationals diagonal, then doesn't that mean they don't pair up 1 to 1?

[hippasosmetapont]

The notion countability has been disproved.
If all positive fractions can be enumerated, then the natural numbers of the first column of the matrix

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
...

can be used to index all fractions (including those of the first column). In short, there is a permutation such that the X's of the first column

XOOOO...
XOOOO...
XOOOO...
XOOOO...
XOOOO...
...

after exchanging them with the O's cover all matrix positions. But this is obviously impossible.

[raghavendrakaushik]

While going diagonally, you should not recount the fractions. For example, 1/1 is the same as 2/2, hence it should be skipped. But there would be still a one to one correspondence between rational numbers and integers.

Also, you could have proved by including negative fractions also.

[Rudra-gous]

i have learnt a lot from this lecture
for you many many thanks

[aa-gcfg]

Sir I am preparing for kset, may I get some guidelines from you please please

[disonymity]

The reasoning is flawed and implies that infinites all are the same size. Measure theory handles this type of problem more appropriately and arrives at the intuitive correct answer that there are more unique rational numbers (between any two values, including negative or positive countable infinity) than there are a proper subset such as natural numbers or its proper subsets smaller even or odd.

[SnowIsSorry]

But why is the argument that set of rational > set of integers on the number line? There is obviously another set of infinities between integers?

[shreyanema]

If something goes upto infinity, then how could we count them??
Just like the very basic natural numbers.... How they are countable????

[schontasm]

Errr no.
Clearly the diagonal argument is wrong. There are n integers and nxn fractions. Hence for any n integers there are at least n rationals. This holds for all n including n=infinity.

Hence there are infinitely more rationals than positive integers.

Ditto primes and integers.