=Will somebody care to explain how "0.99999999..." is "1" in the real world and under what circumstances?
I am sorry I brought that up, it was perhaps off topic, I'll address it below never the less.
Now your example shows that you are a little confused about math, let me explain:
Also, in the real world there is never a speed of 1m/sec. If you have to be precise, and in this case we must be, there can be 0.999999...m/sec or 1.0000... m/sec but not 1m/sec.
You are right in the first. We can only make measurements to a certain degree of accuracy, and objects in the real world are subject to countless constraints, so the speed of the fly would never be constant, and it would rarely be exactly 1m/s.
The next part of your statement seems to be a little confused though.
When you write 0.99999... it is a certain kind of mathematical notation, that means that the infintely repeating fraction defined as
9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...
Why do you state that the fly cannot have the speed of 1 m/s but it can have the speed of 0.9999... m/s?
Now to the fact that 0.999.. = 1
remember that when I write 0.9999... what I mean is
9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...
That is the infinite sum of the factor written above.
A basic fact of real numbers, is that if you add real numbers you get a real number. A more advanced fact is that if you add an infinite real numbers the result is either undefined, infinite or a real number.
So the number 0.9999... is a sum of infinitely series of real numbers.
Lets on a wild hunch guess that the sum of this series is 1!
We do not know this, but when we have a guess we can test if we are right.
We do this by looking at the limit of the sum
9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...
Now if the sum is different than 1 then there must be a difference between the sum, and 1. That is
1-9/10 + 9/100 + 9/1000 + ... + 9/10^n + ... <> 0
Now lets look at it piece by piece...
1-0.9 = 0.1
1-(0.9 +0.09)=0.01
...
1-(0.9+0.09+...+9/10^n) = 10^(-n)
The last statement shows how the sum of the series relates to the number 1. If you have 10,000 elements in the sum, then the difference between the sum and 1 is 10^(-10,000)
As you can see the difference between 1 and the sum decreases steadily .
This means that the two are the same! Why?
Let Z= 1-0.9999...
Now if 1> 0.9999... then Z>0, this is a basic fact of real numbers.
Assume that Z is a real number greater than zero, that is, assume that 1>0.999....
Now since Z is a real positive number we can always find a N such that 10^(-N) < Z.
That is we can always find a number smaller than Z. This again is a basic fact of real numbers.
But this means that the sum
9/10+9/100+...+9/(10^N) is closer to 1 than Z, or
1-9/10+9/100+...+9/(10^N) < Z
But that is impossible, since we know that
0.999.. > 9/10+9/100+...+9/(10^N)
So assuming that Z is larger than zero gives us a contradiction, we must then acknowledge that
1-0.999.. = 0
or in other words 1=0.999...
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