Just thinking said:
Take for example the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... an infinite number of times, with each succeeding fraction being 1/2 the one prior to it. This series becomes 1 ...
No, it doesn't. Series don't "become" anything. A series is what it is. A series no more "becomes" something than two plus two "becomes" five for large values of two. The
limit of the series is 1. "Limit" is
not what the series "becomes", it is
not what the terms "go to", it is simply a mathematical property of the series.
How can it be 1 if it never reaches 1?, some will say.
And the reason they say that is because they are extending the metaphor too far. This problem can be eliminated by simply abandoning the metaphor rather than going to such great lengths to make the metaphor work.
geni said:
If the person thinks the two are differt finding numbers between are trivial.
No, it's not. Name a number between them. Giving a formula is not the same thing as naming it.
.999...=1 is not a stadard mathatical axiom
It can be derived quite trivially from basic axioms. The whole issue only arises because people have somehow come to the belief that there is some objective meaning to the symbols that can be derived separately from the definitions we have given them.
If you want to work from axioms you use these:
I don't need those. I can simply declare that if a decimal point is followed by n digits, then m digits with a bar over them, then that represents the ratio of the difference between the concatenation of the first sequence of digits and the second sequence minus the first sequence of digits divided by the concatenation of m nines and n zeros.
Or I can declare that a sequence of digits, followed by a decimal point, followed by another sequence of digits, followed by a sequence of digits with a bar over it, is equal to the original number concated with the sequence of digits with a bar over it. The above can then be derived from that.
Really, all you need is for the algebra in the following proof to be valid (I'm going to use underlining in place of overlining):
1) x=.
9
2) .
9=.
99
3) .
99=.9
9
4) x=.9
9
5) 10x=9.
9
6) 10x-x=9.
9-.
9
7) 9x=9.
9-.
9
8) 9x=9.
0
9) 9x=9
10) x=1
Remember, the bar over the numbers (or under, in this case) is a mathematical symbol, and can be defined by mathematicians. There is no need to appeal to external axioms. Just make sure that you have axioms that allow all the above steps. Asking for a proof that .999...=1 is like asking for a proof that the radical symbol represents the square root.
Remeber Fred Richman is a legit professor.
It's irrelevant whether he's a legit professor. What matters is whether
his argument is legit, and I don't see that it is.
geni said:
The problem is you live in the wrong physical reality. Since the universe is not infinitely divisible it is possible (using standard arithmetic)
How do you know that?
