This makes sense. Thanks.
Interestingly, it was obvious that it must converge to 1: Representing the fractions as binary fractions gives you:
{ 0.1, 0.01, 0.001, 0.0001, 0.00001, … }
or, as a sequence of partial sums:
{ 0.1, 0.11, 0.111, 0.1111, 0.11111, … }
which is very similar to 0.999~ in base 10.
1 + 0 = 1
1 * 0 = 0
The sum is one and the product is zero. I’m not sure why you consider this any weirder when it happens for a sequence than when it happens for two numbers.
Everyone seems to have already thoroughly explained why 0.999… = 1. (But I can’t resist adding my 2 cents). I think the key is in understanding that a repeating decimal is defined to be the limit of a sequence.
However, if I could try to put the flaw in the original poster’s reasoning into plain English, I would say this:
He assumes that infinity minus one is not equal to infinity. (So if there were infinity nines to start with, there would be less than infinity in the end.) But this is false, since infinity minus one is still infinity. If you consider that impossible for “numbers”, then it simply means infinity isn’t a number.
This argument isn’t really relevant, though, because really 0.999… = 1 because that’s what it is defined to be. We define repeating decimals to be the limit of sequences, and in this case the corresponding sequence is one whose limit is one. The mistake some people make is in thinking that it is not necessary to have a separate definition for repeating decimals – that they instead can be defined in the same way as terminating decimals. In other words, people think that because 0.237 has a meaning, then the same definition can be used to define 0.3333… But if you think about how you define 0.237, this isn’t the case. To determine what fraction corresponds to 0.237, you count the number of digits after the decimal point, call that number n, and then define this decimal expression as being equal to the fraction whose numerator is the integer that appeared to the right of the decimal point (in this case, 237), and whose denominator is 10^n. Thus, in this case the fraction is 237/1000. My point is that this procedure fails for non-terminating decimals, because you can’t count the number of digits after the decimal place. So a new definition is required to give these expressions meaning, and the definition that mathematicians have agreed to use automatically makes 0.999… = 1.
Well, as covered in an earlier thread it depends what you mean by “infinity”. In Conway’s numbers, the appropriate notion of “infinity” is the first transfinite number “omega”, and omega-1 is actually less than omega. On the other hand, 1-1/omega is “infinitesimally” less than 1 so the “infinitesimally less than” argument might seem to go through. Even so, Conways numbers are a proper ordered extension field of R and so have .999~ = 1.
One question: What is
.999~ + .111~ ?1.111~, or 10/9.
I had to go back to the OP to check, but you said 1/3 = 1/3