Quote: "In the picture ionstream posted, the website even states the repeating decimal is just a decimal "approximation", which I can see causing some confusion."
The answer with "0.333333333333333333333..." is labeled as an approximation because it cannot show infinite 3's. The answer "0.3_" (repeating) is not an approximation.
Mr Handy has sort of lost what he was trying to argue with to begin with. First he thought 0.(3) is less than 1/3rd, then he thought it was above 0.3 because Darkbasic said so, which might be his current stance, I don't know. He is also selectively ignoring our complete decimation of his logic.
Quote: "mr Handy came from dark cave, me don't know what float is. Float is some thing floating on the river? Brain hurts."
If you think a floating point number system can prove anything about math, much less something with recurring decimal notation, then yeah, apparently you don't know what a float is or how it works or how math between two floats works. Your logic is that "computers are meant to compute therefore what my computer spits out is right." This sorta tells me you have no idea what's going on inside your computer. Use a more advanced scientific package, like sympy, Matlab, mathematica, or Wolfram Alpha if you don't believe me.
Quote: "Just because you can't imagine that?"
No, because the notation you're using says it doesn't exist.
Why is 1/3 - 0.3 repeating = 0 with no difference (not remainder, remainders have to do with division)? Because for any given iteration of subtracting 1/3rd by 0.3 repeating, the difference is going to be filled in by the next digit.
First digit:
1/3 - 0.3
= 1/3 - 3/10
= 10/30 - 9/30
= 1/30 <- the difference so far
Second digit:
1/30 - 0.03
= 1/30 - 3/100
= 10/300 - 9/300
= 1/300 <- the difference so far
Third digit:
1/300 - 0.003
= 1/300 - 3/1000
= 10/3000 - 9/3000
= 1/3000 <- the difference so far
xth digit:
1/(3 * 10^x) - .3*(10^-x)
= 1/(3 * 10^x) - 3*(10^-x)/10
= 1/(3 * 10^(x+1))
Because this is true for all digits in 0.3 repeating, the difference in any particular digit is subtracted by the next digit, and the next, and the next. Therefore there is no difference - it is wholly accounted for. If you say, "The difference starts here at this digit, " I can say no it doesn't, its handled by the digit next to it, for any digit you choose.
Bonus, it's another proof for why 0.999... is 1.
Surely you can see the glaring, massive error with this little equation? 3 + 1/3 is your repeating term? As in, 6.3333?
