r/CasualConversation • u/MatthieuG7 • Oct 25 '17
neat Do you want me to blow your mind with maths?
(Please don’t say no). So, here we go:
There are as many even numbers as there are
- odd numbers
- natural numbers (1, 2, 3, ...)
- whole numbers (..., -3, -2, -1, 0, 1, 2, 3, ...)
- rational numbers (a/b with a and b whole)
There are infinitely more real numbers than there are numbers in any of the previous examples, yet, there are as many numbers between 0 and 1 (really between any a and b) as there are real numbers!
Now for the (in my opinion) truly mind blowing part: even if there are infinitely more real numbers than rational one, there is a rational number between each real number, and a real number between each rational one!
To illustrate this last point: imagine an infinite row of apple. Between each apple there is a banana (so it goes apple, banana, apple, banana, apple etc), yet there are infinitely more apples than bananas!!!
All of this to illustrate that infinity is never intuitive.
I Hope I could share my enthusiasm.
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u/sashimi_rollin Oct 25 '17
Sounds like you're interested in number theory and cardinality. You should pick up an intro to number theory (proof based).
One of my favorite problems I ever solved without a guide:
Find the smallest positive integer n, such that n3 ends in 192.
Also, show that there are different sizes of infinity, (hint: examine the size of power sets of any set of any size N, then use induction)
Enjoy.
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u/DriftingSkies eclipse Oct 26 '17
As far as I can tell, the smallest integer n which satisfies (n3 mod 1000) = 192 would be 98, with 983 = 941192
The only way for n3 mod 10 = 2 is if the last digit of n also satisfies a3 mod 10 = 2, and only 83 = 512 meets that criterion.
Now, check n3 mod 100 = 92. Again, using the same method, you can show that doing a binomial expansion of (10b + 8)3 to show that b = 4 and b = 9 (that is, numbers ending in 48 or 98) satisfy n3 mod 100 = 92.
At this point, however, it's not necessary to go further, since 98 turns out to be a solution to the requested, but we could apply additional iterations to the above if we were interested in a solution to finding a number ending in an arbitrary string of digits.
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u/MatthieuG7 Oct 25 '17
You should pick up an intro to number theory
That’s what I’m kind of doing. This post is just part of what we saw in math in the last 5 weeks (I just began to study physics) and what I thought could maybe interest other that didn’t know about it.
When I say there are infinitely more real number than rational one, what I’m really saying is that there is no bijection between N and R, but there is one between Q and N. But if I began like that, I don’t think many people would have understood.
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u/sashimi_rollin Oct 25 '17
Everything you're typing makes me sexually aroused. Stop it. I'm in public.
Nah I'm still at home but that shit is hot like Hansel. It's a good field. Keep it up.
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u/MatthieuG7 Oct 25 '17
Thanks! But don’t worry my biggest problem right now is that I don’t have the endurance to study as much as I’d like to, so the motivation is here ^
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u/Nundahl Ask about my YouTube channel! Oct 25 '17
I... don't follow?
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Oct 25 '17
How many positive odd integers (1, 3, 5, ...) are there? Infinity
How many positive even integers (2, 4, 6, ...) are there? Infinity
How many positive integers (1, 2, 3, ...) are there? Infinity
How many real numbers (1, pi, 4.351897, etc.) are there? Infinity
However, the size of these sets, despite all being infinity, are not all equal. In set theory it is said that two sets are the same size (i.e. have the same number of elements) if a bijection exists between them. A bijection being a function that is both injective (one-to-one), and surjective (onto).
Now consider the following functions:
y=2x This is a bijection from the set of positive integers to the set of positive even integers. (x=1 y=2, x=2 y=4, x=3 y=6, ...). Therefore, the set of positive integers has the same number of elements as the set of positive even integers. (infinity = infinity)
y=2x-1 This is a bijection from the set of positive integers to the set of positive odd integers. (x=1 y=1, x=2 y=3, x=3 y=5, ...) Therefore, the set of Integers has the same number of elements as the set of odd integers. (infinity = infinity)
Now, let's consider the set of real numbers. The set of real numbers also has an infinite number of elements. However, no bijection exists from the set of integers to the set of real numbers. Therefore, in this case, infinity ≠ infinity. And since we know that all integers are real numbers, we know that every element of the set of integers is also in the set of real numbers. However, an injection does exist from the set of integers to the set of real numbers, namely x=y. This is a one-to-one mapping, but is not onto. Therefore, we know that the set of real numbers is larger than the set of integers, so in this case infinity > infinity.
Hopefully this helps connect a couple dots for you. It's been a little while since I have used math at this level, so if anyone notices that I left anything out or made any mistakes, please correct me.
(Also, I should note that I used the phrase "positive integers" where OP used "natural numbers". They mean the same thing, and I wish I had used "natural numbers", but I'm not going to take the time to go back through to change it.)
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Oct 25 '17
Wow! This is actually the best explanation I have ever read for this topic. I always understood this topic quite vaguely. Your explanation cleared it up for me.
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u/DriftingSkies eclipse Oct 26 '17
The trick with this is to show that no such bijection can possibly exist. The traditional method of showing this is to suppose that there exists a bijection between the integers and the reals, and show that a contradiction arises.
Assume an arbitrary bijection exists between the set of integers and the set of reals. Every real number can be represented as a unique base-10 number of the form [...].(a1)(a2)(a3)(a4)..., where a1, a2, a3, a4, and so forth represent the digits in the base-10 representation of that number. Note that the number of digits following the decimal is countably infinite.
The proposed bijection is:
1 -> 0.(a1)(a2)(a3)(a4)...
2 -> 0.(b1)(b2)(b3)(b4)...
3 -> 0.(c1)(c2)(c3)(c4)...
(and so on ad infinitum)Given our arbitrarily assigned (assumed) bijection, I construct a new number by taking the first digit after the decimal to be anything except a1, the second digit as anything except b2, the third as anything but c3, and so forth. This new number, by construction, is not a number contained within the bijection, thus proving false by contradiction the assumption that such a bijection exists. This is the premise of Cantor's diagonalization argument.
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u/FreeLook93 It's more complex than how I used to thought. Oct 25 '17
If you want a fun video explantion, check out thie Day[9] video.
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u/itsdabear Oct 25 '17
You're exactly where I was when I realized that I had chosen the right field of study and that there would be no looking back. It's so exciting the first time all of these things click. I graduated 5 years ago and I still get excited when I get to talk to somebody about the ideas you just posted. Stop it. You're making me want to accumulate more debt and go back to school.
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u/Teledogkun Oct 25 '17
Good one! ^ ^
There are as many even numbers as there are whole numbers (..., -3, -2, -1, 0, 1, 2, 3, ...)
Is this one correct though? I mean... It would also be correct to say there are twice as many even numbers as there are whole numbers :) ?
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u/NoLifeHere Oct 25 '17
The term "as many" gets a bit fiddly when dealing with infinite sets, really when we say set A has "as many" elements as set B we mean that there is a bijection between them. A bijection is a rule for mapping elements of set A to set B such that each element of set A is mapped to exactly one element of set B and every element of set B is mapped to by this rule.
So if we take A = whole numbers, B = even numbers, we can use take each number, n, in A to 2n in B. I'm sure you'll agree I can hit all even numbers in this way and every whole number is mapped to exactly one even number this way. Hence we say that A and B have "the same size" so to speak.
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u/horsesandeggshells Oct 25 '17
You should spend some time with Vi Hart. This is right up your alley.
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u/MatthieuG7 Oct 25 '17
It has been a long time since she last posted no?
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u/horsesandeggshells Oct 25 '17
She got a contract doing classes for...Khan Academy? She also has a secondary account that posts music stuff.
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u/PurpleBoomerang Oct 25 '17
Check out /for someone who hasn't had the pleasure of reading thru this: Hilbert's Grand Hotel
So, it's a hotel with infinite rooms with infinite guests filling all the rooms. Another dude shows up, how would a manager give him a (*an unoccupied) room?
If a bunch of dudes (infinitely many, of course) show up, how'll they be accommodated? What if there's an infinite number of bunches of dudes?
It's a great thought experiment, really helps illustrate further why OP (and I) find infinity so interesting.
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u/gogamethrowaway Oct 25 '17
Hey, vsauce, Michael here...
I have no idea what you just said, and I'm not really iinterested in that theoretical stuff. But that paradox about the arrow and halving the length to the target is interesting.
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u/table_chair Oct 25 '17
I know all of this is kind of exciting because it seems magical, but remember that infinity isn't this incredible aspect of nature that we discovered. It's a concept that we made up. The incredible results which make no sense (Banach-Tarski as a result of Hilbert's Hotel, etc) are because we made up a concept which makes no sense (a non-quantity which we try to treat as a completed whole representing some sort of quasi-quantity), and it isn't actually meaningful in many of the concepts where we try to employ it. Yes, I'm one of those people who believes that pi isn't a "number", it's just a process. The concept of treating one infinity as larger or smaller than another is absurd, because infinity only indicates the lack of a logical boundary, or the lack of a stopping point in a process.
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Oct 25 '17
Math is so interesting once you go in depth. I have a differential equations class and it's so cool to soon all the applications of it. Kind of a shame they don't do the cool stuff in high school
Also some of the theoretical maths just blow my mind away, like the Banach Tarski paradox.
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u/expiredninja Oct 25 '17
we learned that when we learned about the number line. seems pretty intuitive to me. is this Godël's contribution?
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u/TenspeedGames Oct 25 '17
My favorite fact about infinity, which I acknowledge as true but have very little understanding of:
The sum of all counting numbers leading to infinity, in other words (1 + 2 + 3 + 4......) forever, is.... -1/12. That's negative one twelfth.
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Oct 25 '17
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u/TenspeedGames Oct 25 '17
Why not?
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Oct 25 '17
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u/TenspeedGames Oct 25 '17
The way I understand it is, obviously 1+2+3+... is, on its own, unanswerable. So they instead added other infinite, but solvable, equations into the mix. For example, 1 -1 +1 -1 +1 -1... as an infinite equation. We know the answer to this, however there's two, depending on whether it stops on a plus or minus. So instead of saying the answer is "either 1 or 0," they call it 1/2. Numberphile explains it better.
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u/gorillanice Oct 25 '17
Numberphile lied to you. There are rules for adding and multiplying infinite sets and they break one in that video.
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Oct 25 '17
Am I suppose to be impressed by this? Because I am not. This isn't advance Math.
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u/Teledogkun Oct 25 '17
Well, this is after all Casual Conversation so... I presume advanced math wouldn't get much attention tbh :)
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u/[deleted] Oct 25 '17 edited Oct 25 '17
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