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# kastro

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1. ## 1 = 0.999...

Sorry, not meant to offend--but yes, it might be close to mathematical espionage. To be honest I know next to nothing about p-adics; any extensive study of them is probably reserved for an advanced topics grad-level course.
2. ## 1 = 0.999...

Not quite. I'll explain why, but to be honest if you dispute that 0.999... = 1 (and I promise you there isn't a single PhD-holding mathematician in the world that disagrees with this) then you don't really have any right to even know what p-adics are. In any case, 0.999... isn't even 10-adic, since p-adics MUST have a finite set of non-zero integers to the right of the p-adic 'decimal'. But even if this result were valid, it would hold true only for the field of 10-adics; attempting to extend it to the reals would be rediculous.
3. ## 1 = 0.999...

Nmenth's objection to this is that infinite sums cannot be multiplied by anything. This is in general true, and in order to make sense of multiplication on infinite sums we require an additional requirement, that it be convergent. In our case, 0.999... certainly converges.
4. ## 1 = 0.999...

These are equivalent. And it is not. An infinite sum may well converge to a finite number, and hence is not at all comparable to 'infinity' in the unbounded sence. In any case, infinity is not a real number, and operations on it are not defined so multiplying it by anything does not make sense. Again, this is precisely part of the definition of the real numbers. You cannot 'refuse' to acknowledge it. Don't really know where you're going with this. Mathematics isn't physics, and doesn't attempt to explain real-world phenomena. Given any two distinct numbers, the average of the two exists, and lies strictly between them. So basically x was defined to be infinity. It should be obvious why this argument is false. No such 'closest number' exists--there's always a closer one.
5. ## 1 = 0.999...

I'd debunk most of these arguments, but to be honest most don't make a whole lot of sense...
6. ## 1 = 0.999...

Not sure how pi is infinite, as it's certainly bounded above by 4. All these claims that you're making essentially 'break' mathematics, and would imply that the last ~400 years of work in the field has been for nothing, which should tell you immediately that you're probably wrong if for no other reason than the math actually works.
7. ## 1 = 0.999...

pi * 1/pi = 1. There, I multiplied pi.
8. ## 1 = 0.999...

All this comes from a lack of understanding of infinity, and this is natural without a formal study of the matter. For example: Which is bigger, the set of positive whole numbers, or the set of positive whole even numbers? I.e. {1,2,3,...} vs. {2,4,6,...} Which is bigger, the set of all numbers between 0 and 1, or the set of positive whole numbers? etc.
9. ## 1 = 0.999...

I don't have much interest in 1/3 = 0.333... arguments, because the equivalency of this with 0.999... = 1 is so immediate that a failure to understand why the latter is true will surely lead to a dismissal of the former. I address only the following: I don't understand what you mean by "it applies only to numbers ending in 999...", as it's certainly true for all real numbers. 'Why' it's true is fairly sophisticated, but it should be intuitively obvious that given any two distinct numbers a and b, (a+b )/2 exists and certainly lies between them. Regardless, explicit construction of the real numbers is often ignored (all most people need to know is that it can be done), and instead we simply define them as a system of numbers with certain properties, this being one of them. The 1, in fact, does not exist. Furthermore, 0.000...1 is not a well-defined number as stands. The only possible definition it could have is the limit point of {0.1, 0.01, 0.001, 0.0001, ... }, but this sequence clearly has 0 as a limit, and limit points are unique, hence 0.000...1 = 0. If you can think of another logical way to define it then by all means go ahead, but "0 followed by an infinite number of 0's, followed by a 1" is, again, not well-defined. Indeed, you have observed that the limit of {0.9, 0.99, 0.999, .... } is 1. We define 0.999... to be the limit of this sequence. There is no other possible logical definition; it's certainly not greater than its limit, and it cannot be less than it (the important point here), so they must be equal.
10. ## Video of the day

Haha. Just watched him in True Romance (would recommend). Christopher Walken is awesome.
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