is fully evaluated using finite techniques. The key thing to note is that we didn't perform infinitely many operations to get to this point. Mainstream maths therefore considers that, with the appropriate definitions in place, 0.999. I realise that you aren't disputing this. is 1 because the sequence eventually gets closer to 1 than any (non-infinitesimal) margin of error. Formally, for any positive number epsilon, there exists an integer N such that whenever n>N, whichever is positive out of (L - a n) and (a n - L) is less than epsilon. has a limit L if, no matter what margin of error we specify, whatever tiny trillionth part of a Planck length raised to the gazillionth power, the sequence is eventually, if we go on long enough, contained within that distance of L. And we're going to carefully define what we mean by a limit as well. In general (and with a few technical flourishes involving equivalence relations) we're going to define the real numbers to be the limits of sequences of rational numbers. As he's been saying, we need to construct the real numbers.
Their techniques remove the "infinitely many operations" here. VazScep was also talking about how 19th century mathematicians excised infinity and infinitesimals from calculus. VazScep gave a good example of how we can go quickly awry by assuming such infinite expressions behave like finite ones. This is the same distinction as the one pointed out by Thommo and me regarding 1+1+1+., which isn't the same kind of thing as an addition of finitely many numbers. We shouldn't presume that this automatically makes sense or that the rules we use to do arithmetic with rational numbers/finite decimals 1 automatically apply. An infinitely long decimal expansion is not the same kind of thing, for exactly the reason you identify - we encounter this problem of needing to perform infinitely many operations. All finite decimals represent rational numbers in a similar way. A finite decimal consisting of a decimal point followed by n 9s represents the rational number (10 n-1)/10 n.
We need to think clearly about what we mean when we say 0.999. Please don't read anything that follows as a put down. The fact is that decimals are pretty unsatisfactory, which is why they play no significant role in mathematical textbooks on the real numbers. Through analysis, one can determine the limit or asymptote, but that limit can never be reached. JamesSS wrote:I am saying that the ellipsis notation is signifying that a decimal notation can never, ever represent the actual value due to requiring an infinite number of digits.Īny conclusion requiring an infinity of operations, cannot be achieved.
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