41–50 of 50 posts
If the grey were made white instead I would have gotten this immediately.
The point of having three colors is to show that there are three boxes of each size. As such, if you pick any of the colors to represent the series, it's plain to see that all the boxes of that color will make up 1/3 of the total area.
While pedantic, the above "expression" is NOT = 1/3 1/3 is the limit, as the sum of n=1 to n -> infinity, of (1/4)^n The "result" converges towards 1/3. You can get as close to 1/3 as you like, but the result will never quite equal 1/3. Cheers Dion.
I am on the fence about whether this is a good troll or a bad troll. It certainly exploits the "someone on the internet is WRONG" ethos of HN, but I don't think it does so in a particularly amusing way. I'm going to say that it's a rather boring troll. (I do find myself compelled to say that the mathematical convention is that an infinite sum is defined to be equal to the limit of the partial sums, if it exists.)
The convention I have seen is to use the symbol of an arrow such as -> to denote the concept of approaching.
However I did state I was being pedantic.
D.
Earlier quoted context omitted.
Then, I'm afraid, you're missing the point of the paradox. There are a few stories (improperly called "paradoxes") attributed to Zeno: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes You're probably thinking of The Dichotomy. This story points out that matter must not be infinitely divisible. The paired story, The Arrow, shows that a universe of finite, indivisible pieces is also impossible. Thus, Zeno's paradox. Fur…
Also, just because a bunch of puny humans manage to convince each other that the world must be a certain way doesn't mean it actually is.
Earlier quoted context omitted.
I am on the fence about whether this is a good troll or a bad troll. It certainly exploits the "someone on the internet is WRONG" ethos of HN, but I don't think it does so in a particularly amusing way. I'm going to say that it's a rather boring troll. (I do find myself compelled to say that the mathematical convention is that an infinite sum is defined to be equal to the limit of the partial sums, if it exists.)
Well perhaps I delivered the statement poorly. My intent is not to troll, I detest Trolls. The convention I have seen is to use the symbol of an arrow such as -> to denote the concept of approaching. However I did state I was being pedantic. D.
Anyways, assuming that your comment was in earnest, the arrow is typically used for functions (or sequences). E.g.,
1/n -> 0 as n->infinity
1/4 + 1/16 + ... + 1/4^n -> 1/3
1/4 + 1/16 + ... = 1/3
1/4 + 1/16 + ... -> 1/3
I like this! Immediately I can also see that 1/5 + 1/25 + 1/125 + ... = 1/4 To generalize: 1/x + 1/(x x) + 1/(x x*x) + ... = 1(x+1) 'Proved' by looking at a picture :-)
You mean, = 1/(x-1) ;) Can someone please post a non-visual proof of why this is the case? In the meantime, I am working on figuring out my own.
S(n) = 1/n + 1/n^2 + ...
= 1/n ( 1 + 1/n + 1/n^2 + ...) S(n) = 1/n ( 1 + S(n) )
Simple algebra from here:
n * S(n) - S(n) = 1
S(n) = 1 / (n-1)
While pedantic, the above "expression" is NOT = 1/3 1/3 is the limit, as the sum of n=1 to n -> infinity, of (1/4)^n The "result" converges towards 1/3. You can get as close to 1/3 as you like, but the result will never quite equal 1/3. Cheers Dion.
It's exactly like saying lim as x -> 1 of 2x -> 2. You don't write it that way. You write it as lim as x -> 1 of 2x = 2.
A very cute algebraic way to see this is to do arithmetic in base 4. In base 4 the series on the left is: 0.1 + 0.01 + 0.001 + ... = 0.111... (recurring) Multiplying the right hand side by 3 gives 0.3333... = 1, and so the original series must have just been 1/3.
Cute. To illustrate the sum of limits, however, I prefer a graphic with sum -> 1. Take an empty circle then add a chunk of shaded semicircle, then add one quarter, then add one eighth... you get the idea.
A very cute algebraic way to see this is to do arithmetic in base 4. In base 4 the series on the left is: 0.1 + 0.01 + 0.001 + ... = 0.111... (recurring) Multiplying the right hand side by 3 gives 0.3333... = 1, and so the original series must have just been 1/3.
Very nice indeed, but you can do it in binary, where the series is 0.01 + 0.0001 + 0.000001 + ... = 0.010101010101... Multiplying the right hand side by 2 gives 0.101010101010... 0.010101010101... + 0.101010101010... ------------------- 0.111111111111... = 1 = 11*0.010101010101...
1/n + 1/n.n + ... = 0.1111... (in base n)
Multiply by (n-1) to get:
0.(n-1)(n-1)(n-1)... = 1
So the original sum must have been 1/(n-1).
This is not so easy in binary.
Earlier quoted context omitted.
Gah, Zeno's Paradox. A professor tried to stump the class with that one in an introductory philosophy course I took. I then proceeded to introduce him the fundamental principles of calculus with respect to limits. I think I threw in some snark about how this was the difference between mathematicians and philosophers - mathematicians actually find solutions! Then scientists ensure they apply to reality, and engineers…
Then, I'm afraid, you're missing the point of the paradox. There are a few stories (improperly called "paradoxes") attributed to Zeno: http://en.wikipedia.org/wiki/Zeno%27s_paradoxes You're probably thinking of The Dichotomy. This story points out that matter must not be infinitely divisible. The paired story, The Arrow, shows that a universe of finite, indivisible pieces is also impossible. Thus, Zeno's paradox. Fur…
Actually, neither story proves either claim, which is apparent since there are rigorously defined and perfectly self consistent mathematical theories for each case. Philosophers just don't like them because they involve actual mathematical definitions, so they hide their heads in the sand and pretend they don't exist.
From that Wikipedia article, a quote from Russell: Georg Cantor invented a theory of continuity and a theory of infinity which did away with all the old paradoxes upon which philosophers had battened. ... Philosophers met the situation by not reading the authors concerned.
In other words, it's very easy to argue that infinite processes, continuous space, or motion in space don't make any sense if you can't be bothered to learn how they're rigorously defined in the mathematical theory you're arguing about.
And FWIW, philosophy may have been attempting to describe the nature of the universe, but I can't come up with a single example of an actual physical result that's come from the field. You may argue that math and physics sprung from philosophy, but realize that those two disciplines provide most of philosophy's harshest detractors these days.
And as far as retroactive claims that if people had listened to Zeno we might have stumbled upon special relativity earlier? (from the second link) Flat out bull poopy. The "inconsistencies" that special relativity resolves have nothing to do with classical mechanics at all, they have to do with E+M, and without a well tested and reliable E+M theory and the Michelson-Morley experiment to directly show us that the speed of light is constant regardless of motion we would have dismissed relativity theory as far too strange to be true (which, amusingly enough and in spite of massive evidence to the contrary, is a claim I've heard straight from the mouth of a tenured philosophy professor at an Ivy League school). An infinite speed of light would lead to a perfectly valid classical mechanics, albeit one that we could (now) prove is wrong, and there's nothing more logically consistent about either relativity or quantum mechanics that would have led us to either one without strenuous experimentation and Real Science.