I was stunned that this topic should reappear in my life after lying dormant for decades, and then not just once but twice in the last month.
The second time this month was the Straight Dope entry. The first time was in an introductory statistics class discussing just what the purpose of real limits is. The subject of 0.999~ came up as an orthogonal digression regarding whether the upper real limit of the nominally grouped interval from 5 to 7, being 7.5, was in any way different from the lower real limit of the next interval from 8 to 10, such that both mutual exclusion and collective exhaustion within the range could be satisfied.
It is times such as these that I am thankful for the addled nature of my mind that allows me some focus on scmazz like this, when I should be thinking about getting home to my wife or drinking beer or the thousands of other things guys normally pursue.
Anyway, someone else posited that the upper real limit of the interval from 5 to 7 might actually be 7.499~ and the lower limit of the next interval 7.5, ok? Well, I didn’t leap from my seat to go into the equivalence of 7.499~ with 7.5 because the lecturer was having a bad enough time as it was. But I do remember when I was in junior high that we learnt a method to convert repeating decimals into fractional or ratio form thus:
say that the decimal in question is 7.835757~ such that the 57 portion is repeating
if x = 7.835757~ then 100x = 783.5757~ and
(100x – x = 99x) = (783.5757~ - 7.835757~ ) = 775.74, the beauty being the alignment of the repeating portion of the decimal in the minuend with that in the subtrahend in order to remove it from consideration
x = 775.74/99 = 12929/1650 or 7 + 1379/1650
OK, so now if you look at (7.499~ = x) and (74.999~ = 10x), then (9x = 10x – x = 74.999~ -7.499~ = 67.5), and what do you get when you divide 67.5 by 9…why, x = 7 + 1/2 or 7.5, of course.
Zeno was right in a theoritical sense. He wrote this up to lpease mathematicians all over. And boy was he good. Even after centuries and millenia, it facinates mathematicians.
Practically however, we have seen that Achilles will catch up with the Tortoise. How does he do it? He cheats. At least theoritically speaking.
Achillis does not traverse or pass through every points on the race track. He jumps from one point to another which are not adjacent. We all do when we walk or run. Even that Tortoise does that. But Tortoise’ strides are smaller than Achilles and that is why Achilles will overtake Tortoise.
In Zeno’s mathematical world, both Achillis and Tortoise (and whoever cares to go there) will have to traverse every point on the track and they will prove Zeno right. Actually noone gets anywhere in the mathematical world. Perhaps that is why it is the most hated subject in school.
Ralf, Zeno was `wrong’, or at least he lead us into absurdity. Even from a theoretical standpoint, we know that Achilles will reach the tortiose and that the arrow will hit the far wall. We know this due to the Calculus of Variation, developed by Newton and Liebnitz a couple thousand years after Zeno’s time: An infinite sum is equal to its limit, no matter how you choose to represent it.
We also know this due to simple algebra, as laid out so ably by the OP. Read through the labelled steps and realize that 0.99… is exactly equal to 1.000…
Nitpick: You mean the calculus of fluxions, or just plain ol’ calculus. Calc of variations concerns things like optimum paths or shapes, and was developed by the Bernoulli brothers.
Ralf, I must contradict you.
Achilles does not beat the tortoise because he “does not traverse or pass through every points on the race track.”
Instead of thinking about where Achilles feet are on the track, think about where his centre of mass, or his left foot, or his pinky finger or whatever, think about where that is in relation to the finish line. Then he does indeed pass through every point on the track, where the track is the distance from Achilles to the finish.
And still Achilles passes, not because he cheats, but because the sum of an infinite series is not always infinite. This is the wonderfully mind-bending fact that Zeno failed to understand.
Also, Zeno certainly didn’t appease the mathematicians, as I understand it he (and everyone else at the time) thought he disproved the relevance of math.