Zeno’s Paradox(es)

Posted: November 29th, 2007 | Author: | Filed under: Math | 6 Comments »

zeno’s paradoxes are three paradoxes posited by zeno of elea that cast doubt on the idea that there is "plurality and change"(wiki).

here i will deal with a slightly perverted paradox of dichotomy. here’s the setup

"Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on."(wiki)

essentially to resolve this paradox we need to sum an infinite amount of distances to a finite distance. this is a wholly counterintuitive idea but easily yields to calculus.

on with the resolution

we need to sum this set of distances:

88f74bf2cac3059ed901c716386a63cf.png (1)

we can write this as

49187a725b031fbdf04970a17703dcea.png (2)

this read as the sum of all the terms preview59958-0.png as n goes from 0 to infinity. the sigma ( the backwards 3 looking thing, greek s) stands for sum. formally you just put 0 into preview59958-0.png then add it to preview59958-1.png then add it to preview59958-3.png etc. in our problem a=1 and r= (1/2). you should verify that using these values for a and r generates all the terms of our set.

this sequence of numbers summed together is called a series, a geometric series. lucky for us some clever people have already figured out what (2) sums to, or converges to ( more on this in a second). this series with a=1 converges similarly to this one:

equation5.gif (3)

pluging in .5 for r yields S=2. here we need to think a little because we’ve added our set of distances together trying to get 1 but we’ve gotten 2. the caveat is in the index or n values that we used. the first term of the series(=set=sequence) as dictated by formula (3) corresponds to n=0; the first term is 1 because preview59958-2.png=1. now we don’t have a term like that in our set (1) so we simply truncate that term (delete, substract, 2-1=1) and get our desired result; the sum of infinite partitions equals 1. hence zeno is foiled and i can walk to class.

for those that are ever skeptical and remark that i’ve simply woven an equally fantastical tale about sums as zeno has i shall now prove what the geometric series converges to. btw this proof is from wiki of course but is probably due to euclid or newton.

our series looks like this for n terms(later we’ll make n=infinity):

5f2d7a02c1a7bdd55ccd32dc6e9a8d26.png

we can find the formula for the total sum if use a trick. multiply both sides( meaning the left side which is "the sum total" and right side which all n terms) each by (1-r). this will look something like this:

preview59958-4.png

preview59958-5.png (4)

aace2ef45e24c78bd3c3d394f6faf1fa.png

verify this by (4)!

finally

e17add2e3eaa782635408e542ee8be1a.png

which is what we were looking for all along.

now for as n->infinity (read: n goes to infinite {asymptotic logic hint hint})

7a43ba4b952dbddebdf543a0fa2d9b81.png (5)

where lim means limit and the n->infinity below it means… The entire expression is read "the limit of _______ as n goes to infinity".


the most remarkable formula in mathematics

Posted: November 27th, 2007 | Author: | Filed under: Math | 8 Comments »

atleast so said feymnan and he was an insightful guy

here it is:

f4c95151341e4e186cf77437dbc48206.png