When you write a number such as 7.9467, it is a way of representing it. When the numbers are rational, there is no problem, this way of writting numbers is unique, however, when you try to write numbers that are real, this representation is sometimes problematic, since two different writting can represent the same number. In particular, when you write 0.99999... (where ... neans an infinite number of 9's), this represents the same number as writting 1, since according to the decimal notation, 0.9999.... represents the limit of the series sum of 9/10<sup>n</sup> with n = 1 to infinity. the limit of this series is indeed equal to 1.
The 'problem' of a dual representation only occurs for the rationals. Given some base n, there is one and only one representation for an irrational number in base n. A somewhat goofy definition of the rationals is the subset of the reals that have a dual representation in some base.
Oh hai thar fellow miscer. Want me to dig that OTHER thread up? Remember... the one on the other forum where I made sense of the whole 0.999... = 1 "debate?" Please Register or Log in to view the hidden image!
Easy. It's a direct logical consequence of mathematical definition and axioms. Change the definitions and axioms and you might have that 0.999... < 1. Of course, you'll need to define exactly what you mean by that statement. I assume (but don't know) that if you design a mathematical system such that 0.999... < 1, then you cannot possibly have irrational number. This is because in the system we have, irrationals are what one would call "uncountable." If you have 0.999... < 1, then that suggests to me that you would have to make all the numbers countable (how the counting function would look like is beyond me.. but it all depends on how you define your terms and declare your axioms). I sea you...
Why is my laptop .999... and 1? Why can't it just be 1 and then delete .999... from our memory banks because it doesn't even make sense.
There is no "1" in memory. At all. We use "1" to mean something is turned on. The switch is on. Fuck, I can say that the off state is Sqrt(2) and the on state is pi. It's just a representation. And computers have nothing to do with mathematics. At all. (well, as far as how mathematics works) Well really you cannot talk about 0.999999999999999999999999...................... with an infinite number of nines. You actually have to talk about it in terms of limits. But we say 0.999... to mean the exact same thing as \(\lim_{n \to \infty} \sum_{i=1}^{n} \frac{9}{10^{i}}\). Clearly, that expression is equal to 1.
dang it Absane......yes I am talking about .99999...to infinity and NO I am no talking about .999999 limit.
Well then your statements are meaningless from a mathematical standpoint. You're talking about the physical writing of such a number... maybe physical computation with that number... none which can exist unless you replace it with 1.
it makes no sense to you because it cannot be discerned into math? And consciousness also cannot be put into math...so it has no sense to believe in it as well...is it not?
What are you talking about? The whole point of this thread is resolve, from a mathematical standpoint, whether 0.999... = 1. Mathematically, it does. Some people don't want to believe it. You, on the other hand, seem to be talking about something else.... and whatever that is, I now am not sure of. Please, explain.
Yawn. This is been gone over many times already. To put it briefly, 0.999... to infinity is exactly equal to one. In order for two numbers to be equal, there must be no non-zero number that lies between them. There is no number between 0.999... and 1, therefore they are the same number. That's the definition of "equals" in mathematics. If you wish to argue that 0.999... does not equal 1, please tell us what number lies between them. (I’ll give you a hint; there isn’t one).
that whole summation thing assumes that something has an end...that something can be summoned...the whole limit also assumes that something has an end...they even invented a infinity sign as an end...like it is an end. Infinity goes on forever, there is no summation to sum it up...it might have a start but it has no end. I am talking about .9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999......with 9's numerals going to infinity...and that does not equal to 1 because that cannot be summoned in the first place.
