Does 0.999... Equal 1?

[steven787]

draik said:

0.333.... = 1/3
3*.3333= 3*(1/3)= (3*1)/3
0.999.. = 1

Or

0.111... = 1/9
9*0.1111..= 9*(1/9)= (9*1)/9
0.999.... = 1

Or

x= 0.999...
10x = 9.999...
10x - x = 9.999... - 9.999...
9x = 9
x=1
1 = 0.999...

Or

0.999... = 9(1/10) +9(1/10)² + 9(1/10)³ + .... = (9(1/10))/(1-(1/10))=1

 

I still insist, that in any practical application of numbers in the real world:

First Situation (Used in Economics, Most Physics, Statistics)

.(3) ≠ 1/3 Correct

.(3) ≈ 1/3 Correct

.(3) = 1/3 Incorrect

.(9) ≈ 1 Correct

.(9) = 1 Incorrect

OR Second Situation (Used in education, some physics, some theoretical graphing)

.(3) = 1/3 Correct

.(3) ≠ .(3) Correct

 

There's a big difference between '=' and '≈'.

The only time '=' is an acceptable substitute when simplifying for the sake of education, for theoretical graphing, or when clearly marking .(3) as = to 1/3 and not to the value associated with the real number .(3).

In that last situation, .(3)*3 = 1 ≠ .(9) because .(3) is defined as 1/3.

This ambiguity has been discussed at length and only some extreme theorists and smart ass students pretend not to know the difference between .(3) as a real number and .(3) as a representation of 1/3.

If anything this draiks proofs just prove the flaws of ignoring the recognized ambiguity and the problem with using "close values" instead of exact values.

 

Lastly, using scientific notation [Σ(3*(1/10)^n)] is the same as saying .(3).  Proofing with that notation doesn't make .(9) any closer to 1.