Line Segment - Thought Experiment

[BAndrew]

Imagine a line segment of 1 unit(s) long (meters,feet,whatever you want).

You remove 1/3 of the segment located in the middle and you are left with 1/3 in the left and 1/3 in the right. Do this again for each of the two pieces of the segment left. Now you should have 4 smaller segments. Continue this process infinitely many times (I mean imagine it). The process is illustrated bellow:



The thing is. You obviously have removed infinetely many points (points are considered 0 dimensional), but how many points are left? Moreover, how much length of the original segment is left in the end?

[Froge]

1 - (1/3) - 2(1/9) - 4(1/27) - 8(1/81)....
= 1 - (2^0)(1/3) - (2^1)(1/3^2) - (2^2)(1/3^3) - (2^3)(1/3^4) - ....
= 1 - Sigma (n=0 -> inf) 2^n / (3^n+1)
= 1 - Sigma (n=0 -> inf) (1/3) (2/3)^n

Consider just the Sigma for a moment. The formula for summation of infinite geometric series is Sum = u1 / (1 - r) where u1 is the initial term and r is the ratio between each terms.

In this series, u1 = 1/3 and r = 2/3, so Sum = (1/3) / (1 - 2/3) = (1/3) / (1/3) = 1

Substitution back into the equation, the the answer is 1 - 1 = 0

None of the segment is left.

[BAndrew]

@Froge

WOW! I actually wasn't expecting that response, but you are absolutely right! This is the so called cantor set.

Cantor Set

[Froge]

free reps pls

[Statyk]

I was skimming the Cantor Set page and this guy's name just killed me: "Henry John Stephen Smith"

It's like... How generic can your name get? lol

(no offense, I just find that funny)