Cec wrote:rep'nA wrote:Cec,

You've missed the point. Where do you put the first 1 in 0.000000000000001? Well, you put it in the 15th place after the decimal, which means you think 1-(.999...) = 10^{-15}, which is clearly not true........"

Hi rep'nA. Pardon my ignorance of maths terminology but I can't follow your explanation. What does 10^{-15} mean?.

10^{-15} = .000000000000001Cec wrote:Looking at my previous explanation another way what I was trying to explain was this....

1 minus 0.9 = 0.1 No problem here so let's take it further..

1 minus 0.99 = 0.01 Taking this further gives...

1 minus 0.999 = 0.001 and so on.

In my previous post I randomly chose the fifteenth step to show that 1 minus 0.999999999999999 still produces an answer which is greater than zero.

Continuing this exercise will always give an answer albeit decreasing in value but never reaching zero because 1 will always be higher than 0.9999repeating.

Cec

I tried to anticipate this response when I wrote:

rep'nA wrote:Of course that isn't what you meant anyway. What you meant was that the difference is 0.000...1 where there are an infinite number of 0's before the 1. But what does that even mean? If there is a 1 somewhere, in must be the k-th place for some natural number k, that is the difference is 10^{-k}.

The key point in your argument is when you said, "continuing this exercise...". For your argument to be correct, you are only allowed to continue the exercise a finite number of times, placing a finite number of

9's after the decimal point. As soon as you allow an infinite number of

9's, all bets are off.

Early in the thread, Bigtone53 referred to Zeno. His argument (used to prove that motion is impossible) goes similar to your argument. To get from point

A to point

B, you must first travel half of the distance. To get from the

1/2-way point to point

B, you again must first travel half of the distance. To get from the

3/4-way point to point

B, you must (once again) travel half of the distance...So you proceed along the numbers

0, 1/2, 3/4, 7/8, 15/16, 31/32,... You can get as close to

1 as you like, but never actually get there. The best you can do is

(2^n-1)/2^n for arbitrarily large values of

n. Consequently, you can never move from point

A to point

B (and so motion is impossible). Zeno's paradox fails for the same reason your argument is incorrect, our intuition about the finite is just plain wrong when we talk about the infinite.