0.99999.....=1

If you're going to divide that string into lengths of 1/10^n, there will be 10^n pieces. The total length is 10^n * 1/10^n = 1 for all n.

The contradiction I am talking about occurs when n=∞. In otherwords when there are infinite cuts in the string there will be infinite pieces of length: lim n->∞ (1/10^n). The limit gives us: lim n->∞ (1/10^n) = 0. So the infinite summation of those lengths is ∑0 = 0+0+0+0... = 0. Which is not the original length or as you put it "10^n/10^n = 1". Zero does not equal one. This is a contradiction.

This idea of taking the limit as n->∞ is exactly the same thing that goes on when we say .999... = 1 - lim n->∞ (1/10^n) = 1 - 0 = 1.

Yup.



Good on ya! You are correct, 1 <> .9 .

In fact, I can write it out. 1 - .9 = .1

Now, can you write out

1 - and tell me what the result is?

If you have an infinitely long piece of string, and you cut it in half, you still have two infinitely long pieces of string....

(The difficulty, of course, is finding the half-way point.)

Ok? But we can still cut in half a finite string with a length for instance of 1. Right? Each piece would be roughly 1/2.

So far, so good....

U choose to give a liberal answer (i.e., an inexact answer, a false answer)
because of your difficulty in rendering a correct one.
From that, I dissent; find a bridge to sell.



The result is less than 1.
That 's what the result is.

I'm sorry, I can't accept an inexact, "liberal" answer

I fail to see the purpose in your new direction of thought. The length of the string was never infinite. Yet apparently I am supposed to get some insight from that idea. I do not. If there is something more that I should see which you maybe implying by "So far, so good..." you will have to explain to me what it is. Some kind of fault in my reasoning? I do not know.

What you want is the limit, as n approaches infinity, of the number of pieces times the size of the pieces. That is expressed like this:
lim n->∞ (10^n / 10^n)
and it equals one.

I think the expression you just wrote is mathematically valid: lim n->∞ (10^n / 10^n) = 1. But it is also mathematically valid to add the lengths of all the individual pieces together. Each piece has a length: lim n->∞ (1/10^n) = 0. There are n=∞ pieces of string. Now take the sum (i = 1 to ∞) ∑ [lim n->∞ (1/10^n)] = (i = 1 to ∞) ∑ 0 = 0+0+0+0+0... = 0.

If we define the quantity: lim n->∞ (1/10^n) to be equal to zero in the equation .999... = 1 - lim n->∞ (1/10^n) = 1 - 0 = 1, then we must also make that quantity equal to zero length when we are cutting the string of length 1 into infinite (lim n->∞ 10^n = ∞) pieces. If both methods are mathematically valid then why does one method give 0 and the expression you have above give 1. That is a contradiction.

It's not a contradiction. I'm computing the length correctly, and you're not.

The function that represents the length is 10^n / 10^n. If you want to find out the limit of that as n approaches infinity, you don't separate out the denominator, call its limit zero, and then say an infinite number of zeros is zero.

What is lim n->∞ ((2^n) / n)?

The correct result is infinity -- 2^n grows at a much faster rate than n.
Using your method, we'd say it is zero because lim n->∞ (1/n) is zero, and an infinite number of zeros is zero.

lim n->∞(f(x)/g(x)) != lim n->∞(f(x)) / lim n->∞(g(x)) in all cases

So we can no longer use addition to add the lengths together. We can do so for n=1. We get 10 pieces each of length 1/10.
1/10 + 1/10 +1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 + 1/10 = 1.
But for some reason I am not allowed to add up an infinite number of pieces of the length: lim n->∞ (1/10 ^n).
lim n->∞ (1/10 ^n) + lim n->∞ (1/10 ^n) + lim n->∞ (1/10 ^n) + lim n->∞ (1/10 ^n) + ... = 0 + 0 + 0 + 0 + ... = 0.
What exactly is your problem with addition?

You're computing lim n->∞ (n / 10 ^n), and that is zero.

Think about this. Do you really think you can add an infinite number of zeros together and get a value greater than zero? So it would not be a problem with the addition. It is a problem with equating the limit: lim n->∞ (1/n) with 0 that is the problem. No number of zeros can ever be anything.

No. The problem is your assumption that lim n->∞(f(x)/g(x)) = lim n->∞(f(x)) / lim n->∞(g(x)).

If you disagree with lim n->∞ (1/n) = 0, then we might as well end the discussion until you've completed (and passed) a first course in calculus.

No the problem is your assumption that adding zeros together can get you something more than zero. Even if you add an infinite number of zeros. Or even if you add lim n->∞ 10^n zeros together. You think the total equals 1?

You cannot concede anything even when it is obvious. The sum of zeros is zero. The sum of infinite zeros is zero. The sum of lim n->∞ 10^n zeros is zero. Whether you can use math notation or not, there comes a time when you have to use your brain. Nothing combined with more nothing can never amount to something, no matter what. And truly intelligent people understand that.

No. I am taking the infinite summation: lim n->∞, (i = 1 to 10^n) ∑[lim n->∞ 1/10^n]. I am summing the lengths of all the individual pieces that result when I cut a string of length 1 into lim n->∞ 10^n pieces. The lengths of the individual pieces are therefore lim n->∞ 1/10^n = 0 in length by the limit law.

Do you have a problem with cutting a string into pieces?

Do you have a problem with taking a limit to find the length of individual pieces?

Now do you have a problem with using that length and using addition to find the total length of the string?



I have had 2 years of Calculus/Differential Equations. I got a high A in every class.

As for the lim n-> ∞ (1/n) = 0, I find the lim n-> ∞ (1/10^n) = 0 in my arguments for contradiction. So I do know exactly how to do limits. I just do not accept everything that is written in a book at first glance. But I do not take lightly the efforts that mathematicians have given over many centuries. So when I see something like this where there is clearly a contradiction, you can trust that I have thought about it a long time.

"It is a problem with equating the limit: lim n->∞ (1/n) with 0 that is the problem."

Actually, it's a problem with your application of limits, but since you're an intelligent person who understands all of this...

If lim n->∞ (1/n) is not zero, then what is it?

The problem is that you're running into a contradiction and assuming that lim n->∞ 1/10^n = 0 is what's wrong. I'm not saying that every printed word is correct, but if every modern textbook tells you that the limit is zero, then you might want to start questioning some of your other assumptions/techniques...