Didn’t you just answer your own question?
This statement totally ignores the point being made by the poster to whom you’re responding.
The point is you don’t have to go through such questions which really have no definite answer. It’s a bit like “how many angels can dance on the head of a pin?”: you can give it answer, but can you prove it? Instead you say in this theory, in this logic system such a question has no answer, but it is useful to use this definition.
My tax return computer program, the largest national one, finalized my return this year by saying that my balance with the government was -$800. I filed, thinking that I was entitled to a refund of $800, then was advised by the IRS that I was in arrears. I’ve yet to hear a satisfactory explanation as to why, when I own the government a negative amount, I have to send them a check.
Can you be more specific, please? Obviously I got you wrong before already. :o
The question to clarify was, why is ‘performing the infinite series of additions’ not equivalent to the notation 0.999…? I have heard this so many times, but I never got a good answer to that.
Because sums like a[sub]1[/sub] + a[sub]2[/sub] + a[sub]3[/sub]… a[sub]n[/sub] are defined only for finitely many n by the axioms of the real numbers. It’s really that simple. Of course however we do define summation on infinite series by using the definition of the limit I describe above of the sequence of partial sums, but as you can see that definition never actually involves summing all the terms in the same sense as a sum of finitely many terms.
That is to say that if you dig down deep enough in how the real numbers are set up you will find a rule that says “you can do this only finitely many times” and that rule applies to (normal) summation.
Think of it as being like mathematical induction. Suppose I want to prove a proposition P for all numbers. I show that P is true for zero, and then that if P is true for n, then it is also true for n+1.
At that point, the magic carpet snaps out to infinity, and P is true for all numbers. No one demands that I actually count up. “First, you have to prove that it’s true for 1,287,334 and then you have to prove it’s true for 1,287,335 and then…” No: it’s true for all numbers, without the need for a “construction” to obtain any given number.
Same with 0.333~. We don’t have to go through the “process” of adding more 3s. They’re all there already.
[nudges with foot] … I guess 'twas a good thing this horse was dead before y’all started beating it to death.
So, you think it is an invalid approach to “construct” 0.999… by the infinite process below?
t = 0: write 0.9
t = 0.5: append another 9 (0.99)
t = 0.75: append another 9 (0.999)
…
at t = 1 we have written 0.999…
Why?
To make it even more obvious:
t = 0: I move my pen from point 0 to point 0.9 of the number line
t = 0.9: I move my pen from point 0.99 to point 0.999 of the number line
t = 0.99: I move my pen from point 0.999 to point 0.9999 of the number line
…
t >= 1: which point of the number line my pen is pointing to?
With every whack we give him, he decreases to half his previous thickness…
I feel we are now at the point where we are dealing with the greasy spot on the pavement where the dead horse used to be, but maybe that’s just me.
If you are willing to accept that the limit as n approaches infinity of the sum from 1 to n of the series 9 x 10^(-1) is equal to 1, then for t >=1 you must accept that your pen will have come to a screeching halt at 1 at time t = 0.9[\u] and remains there forever for all t > 1.
But please keep in mind that your need to be able to “see” your imaginary pen pointing at an exact spot on an imaginary concept like a real number line is in no way a proof that the notation 0.9 denotes an infinitely repeating decimal that is equal to 1.
Once again. Zeno understood this. He knew that the end point must be exactly one. He knew this because the question is equivalent to starting with 1 and creating an algorithm for building it.
You can do this by starting with 1/2 and then taking 1/2 of each remaining piece.
You can do this by starting with 2/3 and then taking 2/3 of each remaining piece.
You can do this by starting with 3/4 and then taking 3/4 of each remaining piece.
You can do this by starting with 4/5 and then taking 4/5 of each remaining piece.
You can do this by starting with 5/6 and then taking 5/6 of each remaining piece.
You can do this by starting with 6/7 and then taking 6/7 of each remaining piece.
You can do this by starting with 7/8 and then taking 7/8 of each remaining piece.
You can do this by starting with 8/9 and then taking 8/9 of each remaining piece.
You can do this by starting with 9/10 and then taking 9/10 of each remaining piece.
And there’s our old friend .99999999999999999999999999999999999~. It is exactly equivalent to Zeno’s dichotomy paradox. (So are my other examples. And you can continue the algorithm infinitely far.) He would have considered someone who said that there must be 0.00000…1 left on the end an idiot because it was obvious that the piece left at the end varied with each way of constructing 1.0. In fact, it can be 1, 2, 3, 4, 5, 6, 7, 8, or 9. (Although he would have thought this in terms other than the decimal system.) The only thing he couldn’t figure out is how a finite number could emerge out of an infinite process.
But we know how. Cantor and others described how.
netzwelter’s point is now obvious because of that work. All the mathematicians in the world understand that and the reasoning behind that. Zeno would have understood it if explained to him in his mathematical language. He was much smarter than those today who deny that mathematics is nothing but definitions, rules, and procedures. All that matters today is finding a way to express the obvious in non-formal terms for non-mathematicians. That seems to be hard. Maybe it’s because mathematicians expect logic and sense to take precedence even when others steadfastly deny their existence.
My formal math is long behind me. But I can read what others write. This isn’t hard. It’s one of the easiest concepts to get in math. The math part of the math is somewhat more formidable. Neither half is at all arguable.
Yes, I am willing to accept that the limit as n approaches infinity of the sum from 1 to n of the series 9 x 10^(-1) is equal to 1.
But what is wrong with the list of actions taken here:
All the actions taken can be found in the list (there is no action defined for t = 1 and after). In none of the lines of the list I am moving the pen to point 1. So, how does the pen point to point 1 then? Did I really move the whole distance 0.999…? Am I short of 0.999… and 1 after the actions taken on this list?
It didn’t even take Cantor to explain how. Newton had already done everything that needed to be done centuries earlier, and Archimedes had done most of what needed done milennia before that.
The actions taken on the list have all taken place, all infinity of them. They have always taken place. There is no “after” with respect to them.
If you are hung up on needing to know for which value of n the sequence 1 - 10^(-n) becomes equal to 1, then it’s not clear that you understand the meaning of the limit as n approaches infinity.
The actions on the list are taking place between t = 0 and t = 1. t >= 1 is after the actions.