This means that there is no one-to-one mapping between the set of natural numbers and the set of all possible combinations of natural numbers... which is very counter-intuitive.
Is it? All you have is a mapping of infinite elements from A to a single element form B. That's nowhere near a bijection, and is useless in constructing a bijection. Just because there's some kind of a relationship doesn't mean there's a bijection.
What I mean is that it's counter-intuitive after you have fully accepted the fact that, for example, the set of rational numbers is as big as the set of natural numbers (ie. there is a one-to-one relationship between the two.)
As I commented in another post, the first intuition is that of course there are more rational numbers than natural numbers. However, that intuition is wrong. It takes some convincing before one's brain starts accepting that the two sets are of "equal size".
It's thus not strange at all that the same idea could be used to argue that the set of all possible combinations of decimal digits is as large as the set of natural numbers... except that that intuition would once again be wrong. It may be even harder to convince one's brain of this fact, especially after it has been convinced of the bijection between natural and rational numbers.
Scepheo wrote:
I think it might be. Or am I wrong when I say that (0.4999..., 0.9999...) (which maps to 0.4999...) and (0.5, 0.0) (which maps to 0.5) map to the same point?
You might have a point there. (Although I think that 0.9999... is exactly the same as 1.)
I have bad news. Since Kayla has had her internet out or something for about 4 days now, I'm assuming she's gone forever at this point unfortunately, so until she comes back (rather if she comes back at this point in time), she is no longer going to be listed in our organization.
Devastating news indeed. How can we ever cope?
Cake and grief counseling may be needed for all affected.
I think your assertion needs refinement, though: every real number can be represented to arbitrary precision by a finite string of natural numbers. Once you want to represent a number by its "true" decimal expansion, the number of digits necessary blows up too much and you end up unable to represent all the reals.
But I'm not talking about finite representations. What I'm saying is that, if I understand correctly, any real number can be represented completely accurately by an infinite string of digits. Not just as an arbitrarily close approximation, but the exact accurate value.
A consequence of this is that, unintuitively, the set of all possible infinite strings of digits is uncountable (and at least as large as the set of reals.)
This reminds me of a conversation I had with a friend in high school. I had learned enough at that point to know that the digits of pi are "random" (which I now know hasn't actually been proven...) but my friend said that it has to settle down at some point into some kind of repeating decimal or pattern. I said no, it doesn't, but he just couldn't wrap his head around the idea that pi was transcendental.
It's not a question of pi being transcendental. It's just a question of whether it's rational or irrational (ie. a real number that's not rational.) If it were rational, then it would at some point have an infinitely-repeating pattern (that can be expressed as the ratio between two integers.)
The square root of 2 is irrational (and therefore has no infinitely repeating pattern) but not transcendental.
Regarding a question I have asked previously in this thread, namely: "Is the set of all points inside a unit square as big as the set of all points on a unit line?"
If the two sets are equal in size, that means that there exists an unambiguous one-to-one mapping between all the points on the line and all the points in the square. Would this be a valid such mapping?
Take the decimal representation of the x and y coordinates of the point inside the square, and interleave each decimal digit to get a value between 0 and 1 (which would therefore correspond to a value on the unit line).
For example, if the point is (0.1234, 0.5678) then its mapping onto the unit line would be 0.15263748. Or for example (0.12, 0.3456) would map to 0.13240506.
Does that work? Or is it ambiguous (ie. more than one point in the square would map to the same point on the unit line)?
Infinite sets and cardinality are really unintuitive.
At first, it's extremely unintuitive that the set of rational numbers is exactly as big as the set of natural numbers, even though intuition would tell that of course there are enormously more rational numbers than natural ones. Intuition would be wrong.
After understanding the reasoning behind it (mainly, that you can unambiguously map each natural number to a rational number, which makes the sets equal in size), one gets to accept that yes, there are as many. Then the next unintuitive step is to understand why the set of real numbers is genuinely "larger" than the set of rational numbers. (This is quite unintuitive because between any two rational numbers there is an infinite amount of real numbers, and between any two real numbers there is an infinite amount of rational numbers. Intuition would conclude that the sets are therefore comparable in size. Yet somehow there are more real numbers than rational ones. This is quite unintuitive.)
Then, when one finally accepts (at least at some level) that there really are more real numbers than rational numbers (because it can be proven that it's not possible to construct a one-to-one mapping between them), comes the next unintuitive step: Even though the set of real numbers is genuinely larger than the set of natural numbers, you can still represent every single real number with decimal digits (which are, rather obviously, natural numbers.) Intuition fails once again. "But wait, you just said a second ago that there are more real numbers than natural numbers, which means that there is no one-to-one mapping between them. Now you are telling that you can represent every single real number with a string of natural numbers? Isn't that a contradiction?"
Thus the next unintuitive step that one has to understand is that even though the set of natural numbers is countable, the set of all possible combinations of decimal digits is uncountable, just like the set of reals (and therefore reals can be represented with decimal digits)... This means that there is no one-to-one mapping between the set of natural numbers and the set of all possible combinations of natural numbers... which is very counter-intuitive. One would think that you could number each combination uniquely.
I reached the limits of my mathematical knowledge when a question came up whether you can represent all real numbers using decimal notation.
On one hand, the set of reals is uncountable while the set of integers is countable, and therefore it would seem to me that you cannot represent all possible reals with integral digits.
On the other hand, are there real numbers that cannot be represented with integral digits, when we allow an infinite amount of digits? My intuition would say yes, because a countably infinite amount of digits (from a finite set of them) doesn't somehow turn uncountable. Or does it? If it doesn't, then there would be real numbers that cannot be expressed even with infinitely many digits, but if there are, what would they be?
Strangely, I don't know the answer to this, even though I really should.
I was trying to search messages I have posted in the "some math challenges" thread in off-topic, but couldn't figure out a way to do that. I tried to put the thread title in the first search box and my nickname on the second, and checked the "search for all terms" checkbox, assuming that it would then show only the posts made by me on that thread, but that didn't work. (Perhaps it's applying an "or" operation instead, or using only the author field.)
Is there any way of doing this?
Nintendo attempted to ban Super Smash Bros. Melee from being livestreamed on twitch.tv.
I really can't understand what exactly they expected to gain from a stunt like that. (It's stupid, really. Such an event is, in fact, completely free advertisement for that game in particular, and all the games in the series in general, and even for Nintendo's consoles. And they look the gift horse in the mouth...)
Or is this a case of Nintendo's lawyers acting without consulting any higher-ups (or, alternatively, convincing the higher-ups to go forward with this, and the higher-ups being too busy with other things to care too much)?
Is it shorter? Does it mean they won't be cutting episodes anymore in order to fit them into 20 minutes?
(I just can't get over how horrendously stupid the whole thing is. They make an opening song that's significantly longer than the original, and they add a stupid and completely useless live action segment at the end... and for those reasons they have to cut parts of the episodes themselves in order to fit the entire thing into a time slot. How stupid can they be, really? And I thought Japanese were smarter than this.)
ColdStardust, could you sign your posts so that we can tell who's posting whenever you make a post? That would already prevent a lot of confusion. Currently, each of the posts in this thread already confuse me as to whether user 1, 2, or even user 3 is posting.
The point is that we all act as a team because we're all best friends! Representing a team reaching for the same goal. If people hate on us we have each other to back up.
How exactly is creating separate usernames for each one a problem for that?
You don't understand. "Hacking" means typing randomly as fast as possible. Therefore two people typing randomly at the same time can hack twice as fast as one person. It makes perfect sense.
Seriously, though, there's another big problem with this scene: If this is a typical setup, the hacker is attacking a server somewhere in the facility, and that computer is just acting as a terminal communicating with said server. Shutting down the terminal simply stops them from interacting with said server, and thus the hacker will be completely free to hack the server without impediment, because Gibbs just severed their connection and stopped them from dealing with the hacker.
Nice job breaking it, hero.
When I read this thread for the first time, and didn't even know that MLP:FiM exists, I was quite skeptical about the notion of cartoons being made with Flash. Needless to say, this cartoon vanished that prejudice completely, and I became to appreciate not only the quality that's possible to achieve with Flash animation, but also the enormous talent of the people making it.
Anyway, I was watching some anime when I realized one rather big advantage of Flash animation over hand-drawn one. (That's not to say that hand-drawn animation doesn't have its advantages, especially given that the total freedom of drawing whatever you want opens more possibilities for drawing style etc.)
In many long-running series there will often be episodes where the drawing and/or animation quality will be significantly and visibly lower, sometimes to the point that it will look outright hideous. This is because long-running series have limited budgets, and making quality hand-drawn animation requires time and effort, and therefore money, so the budget is distributed so that the most effort is put into the "most important" episodes, while filler episodes get less attention.
The advantage of Flash animation is, however, that the quality is even throughout. Every single episode of MLP:FiM looks equally good in terms of drawing and animation quality.
One of the major hurdles to learn and program in C (and to a large extent C++) is that you don't only need to know how to program in general and how to program in C in particular, but in addition you need to follow strict programming conventions in order to avoid the most typical and easy-to-do mistakes (such as leaking memory, using freed memory, accessing arrays out of boundaries, accessing memory through pointers that are pointing to completely wrong locations, using uninitialized variables...) These conventions are in no way enforced or checked by the programming language or compiler itself.
In most of the so-called higher-level languages it's either impossible to make such mistakes (eg. leak memory or use freed memory) or the runtime environment will always tell you exactly what went wrong (eg. accessing an array out of boundaries) thus not requiring any programming conventions that exist solely for the purpose of avoiding those mistakes.
C++ offers many tools that make it much easier to write safe code without having to follow so many programming conventions, but it isn't as easy as with other languages.
