Why is your brain being mean to my poor posts?robly18 wrote:Man, you sure have great skill at making walls of text my brain hurts when reading.
(You don't have to read this part. Summary: Great reason to say 1/0 is impossible, even though your reason is wrong). Ah, now I see. So you're nailing it down for not following the basic axioms of mathematics. Just saying it wasn't a number wasn't a very precise definition (*cough* Andy *cough*) or very valid argument, but saying it can't be because it violates basic properties of arithmetic seems like a more solid argument. It's just that words like 'number' can be varied in their meaning, and it isn't very obvious or provable (at least I haven't seen it proven) what 'not being a number' would even mean, much less why it would imply the impossibility of taking an operation. Besides, i technically isn't a 'real' number, and you could define it as 'not a number' at all, so that would mean you could never take sqrt(-1), but i still follows basic laws of arithmetic. Saying an operation is impossible due to its violation of the axioms that are the foundations of mathematics makes much more sense.robly18 wrote:As for the not a number thing, here's why it isn't a number:
Once again, may I say you offer a very clear explanation, but as I hope to do the same, you should hopefully see that the logic of these axioms needs to be revised when discussing ∞, and that ∞ actually follows these axioms anyways.robly18 wrote:The axioms apply for everything. They're the basis of mathematics. They are literally what defines our system of counting, adding, subtracting... If something doesn't obey these axioms, then it isn't really a thing.
Actually, n=n+1->0=1 is not exactly correct. You see, when you write the whole thing out, something pops up:robly18 wrote:As for the "how can it not be a number thing" let me make an analogy.
Say, n = n + 1. Is this possible?
As my good old friend Jim would say, this would mean 0 = 1. It's not possible as you can see.
n=n+1
-n -n
n-n=n-n+1
Now, you might say: "Hey! n-n=0, right?" Well, that's right, but remember, we're taking an operation here, and operations can return multiple values. This may hurt your head, but it happens a lot when it comes to ∞, and it happens all the time with square roots, too, so it isn't an entirely new concept. Now, remember, since n=n+1, we can plug in n+1 in as n, and we get this:
n-n=n-(n+1)+1
0=n-n-1+1
0=0+0
0=0
Now, before you start telling me how this violates the basic laws of mathematics, I never said n-n wasn't equal to 0. It just could be equal to other things, too. This doesn't mean 0 isn't one of its 'solutions'. You have to remember that some operations have multiple values, but only one of them is the 'right' one (remember that discussion about 0/0?), and sometimes we can even deduce which one is right, but you can't just pick out one of those values and plug it in, because it'll get contradictory things, like you've shown. In this case, n-n actually can't be 0 for both sides of the equation. It's kind of like derivatives. A derivative is actually just 0/0, but by doing some cool tricks, you can find out which of the values returned by the operation 0/0 was right. n-n is the same way: it's indeterminate, and that means you can't just assume one of it's values is correct.
robly18 wrote:Jim isn't a real person. Anymore, at least.
Remember, 1/0 can be multiple values, and 0*1/0 can be multiple values, too. Think of it this way: You take both square roots of 1, and then you set it equal to both the square roots of 1. -1 is a square root of one, right? So is 1. So shouldn't that mean that -1=1 because they both equal 1^(1/2)? The thing that's wrong here is that you're picking out one value equal to 1^(1/2) and setting it equal to another value equal to 1^(1/2) and saying they're equal, which they aren't. The exact same thing is happening here. Yes, 1/0*0=1, yes 1/0*0=0, but both these equations aren't completely correct. A better answer would be 1/0*0={0,1}. You can then choose an answer based on context (that is, if you have context). Even so, this wouldn't be the best answer. In fact, the best answer is indeterminate, because 1/0*0=0*1/0=0/0*1=0/0, so 1/0*0 is actually 0/0. You can argue 0/0 isn't a number either, because n/n=1, but because it has a 0 in the numerator, it's the same as 0*1/0, and therefore should be 0 because 0*n=0, but we call it indeterminate, NOT undefined, and it's because it actually equals both/either of these numbers, and we should at least give some such name to 1/0, be it (which could even stand for a set of numbers if 1/0 in fact returns more than one value) or whatever.robly18 wrote:Anyway, as for the n = 1/0, here's why it's wrong:
Multiply both sides by zero
0n = 1
Use axiom proofs to figure out this:
0n = 0
And then ,you get 0 = 1.
As I have shown, there may not be a number equal to 1/0, but that doesn't mean there are no numbers equal to 1/0. Also, your disproof of a number equal to itself plus one overlooked the possibility that n-n could have more than one possible value. Finally, I would like to introduce the notion that the axioms aren't even extendable to this realm. The notion that 0 of anything is 0 creates an exception. Just make up a number, , where 0 * = 1, and there's no reason why this number can't be. It would be like saying there's no number that is the sqrt(-1), because it just can't be. Well, you know what we did? We created a number that was the sqrt(-1), and the applications of the number i have not ceased to amaze us. You see, this type of thinking limits us, and as I have shown, limits are deceiving. Axioms are more of a general rule. You can always make up your own exceptions. The logic that created them always has loopholes. Like the 0*n=0 logic forgets that infinite is so large that one thing out of it is literally nothing compared to infinite, so therefore, that one thing would be 0 of infinite, which although it 'violates' a 'fundamental axiom of mathematics,' has a logic of its own. And logic is what mathematics was based on, not some set of axioms created by mathematicians centuries ago.robly18 wrote:Therefore, the existence of a number that is equal to one over zero is impossible, just like the existence of a number equal to itself plus one. They do not obey the axioms and cause inconsistencies.
You know what we've been doing all these years? Sweeping it under the rug, and now it's time for it to come out.robly18 wrote:It may sound like a case of "it doesn't obey what we know, so sweep it under the rug", but that's the thing.
So the solutions to inconsistency is to create inconsistency? Their 'nonexistence' makes math inconsistent. Also math is never completely consistent, because it's about logic, not consistence.robly18 wrote:Their existence would make mathematics inconsistent.
But with logic, it can.robly18 wrote:We call the undefined because, with our set of axioms, they cannot be defined.