and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. Rejecting this is tantamount to rejecting logic. Students of mathematics often reject the equality of 0.999. There are a bazillion ways to prove this result. Moreover: None of the existing approaches can describe a fair lottery on a standard infinite sample space (of any cardinality) in a regular way. This can be proven from a definition of decimals that's basically an epsilon delta definition. Infinitesimals are sometimes used informally in physics, but its easy to get the rules for them wrong. Infinitesimals and probability Observe: None of the existing approaches can describe a fair lottery on or (or any countably infinite sample space). and 1 are NOT the same (A great explanation of this paradox is here. is not the sequence of partial sums, or any term in that sequence. However, in the system of numbers known as hyperreals, a system which includes infinitesimals, there exists an infinitesimal number between 0.999. That sequence never equals 1, you are correct. In nonstandard analysis, there are positive infinitesimals, so consider 1-inf where inf is an infinitesimal. No matter how many 9s you add, it will never equal 1, similar to no matter how many times you divide 1 by 9, you will never reach zero.What you described here is a sequence of partial sums. Each time you add another 9, you have gotten 9/10 of the way to making it equal 1 from where you just where, but it still doesn't equal one. It is an approaching infinity number of 9s. So there are an equal number of nines in 0.999 and 9. If we somehow remove a nine from this sequence, then we would still have an infinite number of nines. Its a result of our inability to properly express infinity, so we have worked math around that inability and have come to be able to 'basically' express it. This time, the confusion arises from not grasping infinity. We could have stuck with fluxions infinitesimals' Missing the point that nonstandard analysis is actually pretty complicated to make the whole thing work, and that its absolutely a 20thC idea inspired by the sort of intuitive handwavy original version of calculus, but not really a direct development of that.
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