Zeno's Paradox.

[evrenn]

I think Anton ran this by me once.

"If you want to get to another point, technically it's impossible since you need to go half that distance first then half of the remaining distance and this continues an infinite amount of times." Or words to that effect. It's actually called Zeno's Paradox.

Of course, it's not true. You can easily get to point B

Comments

[mechashadow]

I never understood that sort of thing, and .999999.... still seems to me like it shouldn't equal 1. Closer and closer and closer, but never quite it.

It works with any fraction, right? 1/3, then 1/9, then 1/27th. . .

[evrenn]

Well, there's sort of a mathematical convention that says .999999... = 1. The simplest proof is: since 1/3 = .3333... then 3*.3333 = .9999. But 3*1/3 = 1, so .9999 = 1. It's not saying that .99999 is an approximation of 1 but rather that it's just a longer way of writing 1.

Actually, 1/3 + 1/9 + 1/27 ... + 1/3^n = 1/2

Consider 1/3 + 1/9 = 4/9
And then 4/9 + 1/27 = 13/27
13/27 + 1/81 = 40/81
which is getting closer and closer to 1/2

1/4 + 1/16 + 1/64 + 1/256 ... = 1/3
1/5 + 1/25 + 1/125 + 1/625 ... = 1/4

And I think you can see the pattern here.

[chemicats]

you forgot to keep track of your ellipses("...") when making your argument, if you keep ellipeses in account properly you should see why its an approximation,

the convention to round is an approximation convention, it does not indicate an equal.

"The simplest proof is: since 1/3 = .3333... then 3*.3333 = .9999. But 3*1/3 = 1, so .9999 = 1."

also review "limits"
its sort of like thinking of a discontinuity, it approaches but never arrives like a hole in a function

[chemicats]

oh i forgot to explicitly include how the ellipeses would change things, and that the convention is still ONLY A ROUNDING APPROXIMATION and not an equal at all

The counter simplest proof to your argument is:
since 1/3 ~ .3333...
(.3333... ~ 1/3)
then multiplying through by
3(.3333...) ~ 3(1/3)
then simplifying
.9999... ~ 1

this has alot to do with limits which are one of the main basics of calc... maybe why you didn't do so well in calc

and traveling from point to point, in pure mathematics that is not possible by the series you describe.
in a non mathematical context you could argue that because colloqiually locational "points" can be defined over relatively miniscule areas but are comparatively larger than what would be truly defined as a purely mathematical point in terms of graphing