I just could not hold myself here. Wow, amazing thread!
And a long reply, so read it, if interested
I agree with Vissin 100.1111...%. Absolutely right. One of the best explainations!!!!
Quote:
VISSIN:
0.999... = 3 * 0.333... = 3 * 1/3 = 1
ie. 0.999... = 1
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What we prove is only to our abilities in science, we prove what we can perceive and measure. 1000 years ago we thought earth was flat till science proved otherwise. Anyway...
0.9999...inf. will equal to one depending on the perception of "0.99999...inf".
EXAMPLE: Radius of earth is 6,378.1 km or 6378.1 * 1000 * 100 * 10 mm= 6,378,100,000 mm so 6,378,100,000.99999... will also be 6378.1 km meaning we can accept it is equal to one when we are measuring a planet.
For some calculations we might need more accuracy and there we do not round it off till some numbers.
Depending on our use we can say 0.999999...inf is 1.
And biggie:
Quote:
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In my opinion, the moment you put an end to the series 0.999999......9... , it ceases to be equal to one.
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I am hoping you dont mean this practically (theoretically maybe) because our cities and life is BUILT on these "inaccuracies" or approximations.
We dont say the car traveled 999.999998 meters but 1 km. We CAN say the ant traveled 999.999 meters OR 99999.90 cm
because we MIGHT need more accuracy there.
I think Heisenberg's uncertainty principle explains it a bit too. May get confusing but keep reading it to understand.
Quote:
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...locating a particle (very accurately) (0.9999999...inf) in a small region of space makes the momentum of the particle uncertain; and conversely, measuring the momentum of a particle precisely makes the position uncertain (1 maybe? *smiles*).
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On a side note I think we confuse INFINITE with "never ending, forever", which does not exist except the universe. INFINITE in science/math (I think) is used to say that in a particular case the value may be
too much for consideration. For example
Canada is "infinitely big" for a
snail, because it will never traverse through east to west in its
lifetime. Same way "Limits" in mathematics is for calculations where "approaching 0" means we are happy with it even if it is not, does not make a difference. If we need a zero then we recalculate.