Previous  1  2  3  Next

k i s s i n g

   

Ok heres a one for all you brains. We all know their is a speed of light but what is the speed of Dark?

   

Good answer Blue_Nose, however that's math notation that wasn't covered until first year calculus when I went. I believe today it's covered in grade 12 pre-calculus. Those here who didn't go to university won't recognize limits or sumation notation.

There is another glitch to the problem: no matter how you fold paper you cannot fold it more than 7 times. Even without that limit, the problem involves folding half so to "get to the end" you would have to fold it an infinite number of times. If each fold takes a finite time, it would take infinite time to do that. So no, you wouldn't ever "get to the end".

How far would you get? The distance remaining is 1/2 after the first fold, 1/4 for the second fold, 1/8 for the third fold, so the distance remaining after i folds is 1/(2^i), where i=7 you get 1/128. Every computer nerd has the powers of 2 memorized, right? So a 20cm piece of paper would be folded to 1/128 * 20cm = 0.15625cm or 156.25mm.

If you want to measure how far you went, the first fold takes you 1/2 of the way, the second fold takes you 1/4 of the way, etc so the total is:
sum 1/2 + 1/4 + 1/8 + ...
which is represented as
sum 1/2^1 + 1/2^2 + 1/2^3 + ...
which is more genarally written as
for all values i from 1 to 7, sum 1/2^i
which is written in mathematical notation as

   

Thanks Winnipegger, I've taken so many math courses they all blend together...

Not that it's an important piece of information, but the summation involves adding up the expression while changing a variable. i=1 below the sigma sign means that the variable being changed in the sum is 'i' (as opposed to 'L', or any other variable), and that it starts at a value of 1 - in effect, "i=1" means "i starts out equal to 1".

Since it got lost to the other page, I'll pose the original form of the paradox again, for the sake of discussion:

To walk from one end of a room to the other, you have to start out by walking halfway. Walking half the distance to the wall again, you're now 3/4 of the way there. Walk half of the remaining distance again and you're 7/8 of the way there, etc etc etc.

Each 'half of the remaining distance' gets smaller and smaller and smaller, all the way to an infintesimally small distance remaining. This 'halving' method of walking requires an infinite number of steps, so how can we logically complete an infinite number of anything in a finite amount of time?

A related math 'quirk':

We all (should) know that 1/3 in decimal form is 0.333333333..... repeating forever.

If we multiply that by three, in fraction form we get

3 * 1/3 = 1

but in decimal form, we get 3 * 0.333333 = 0.9999999....

Does that mean that 0.99999999 (repeating forever) = 1 ?

   

This is where math meets logic. This problem was presented to us in algebra class as how not to formulate a problem. The correct solution is to calculate the distance you travel over time, divide the total distance by that speed and you get the time it takes. This having thing just doesn't work.

As for one third, the exact representation has an infinite number of threes. You can represent that by placing a bar over the last three. Multiplying that fraction by three does not produce an infinite number of nines, if you understand limits it will tell you that you end up with one.

I went to lunch with a friend and mentioned this thread. He pointed out that a TV program called MythBusters examined this. They found if you roll the paper under a steamroller (or modern road roller) after the 7th fold you can fold it again, and again, and again, for a total of 10 folds. However, once the folded paper is thicker than it is long it won't fold anymore. The total number of folds may depend on how thin the paper is, so vellum may fold more times than bond paper. But we're talking about a very small difference in the number of folds.

   

One of my math teachers spent over an hour trying to explain that "quirk" was the reason the universe was circular.

   

Most of my girlfriends dad's had chainsaws.

   

Shouldn't it be 2 ^n?

2^i=2^1=2

L/2=L is a false statement

   

It's not a false statement if you understand limits and series notation.

I'm not explaining it again.

   

Well, 0.3 x 3 = 0.9, 0.03 x 3 = 0.09, all the way down the line

Whatever decimal place a 3 is in, multiplying it by 3 is always going to give 9.

The answer is that, despite the perception, 0.99999... is equal to 1.

The long trail of infinite 9s is troubling, but it's really no different from the long trail of 0s that we take for granted in any terminating number.

   

The logic is sound, but that does not mean the conclusion is accurate. First can human being understand math. Second can human beings comprehend paper being halved ad infinitum. Finally can all events in the universe be expressed mathematically?

Pythagoras argued that all events can be expressed mathematically but much of the mathematics was inaccurate, it was the work of later philosophers mathematicians and physicists to create many of our mathematic expressions. It seems that paper can not be spilt ad infinitum on account of particle theory. As for the equation itself, if matter is energy, and energy is always lost, then it is only logical that matter is always lost. If you add a subtraction to counter the result will eventually equal 0.

   

What conclusion? That 0.9999....=1, or that a particular infinite series of measurements converges? What's innaccurate about it? It certainly wasn't me who decided either was so (they're based on the same concept). By all means, present your 'innaccuracy' case.
Yes, yes, yes.
That our knowledge adapts to new information over time is hardly a case against knowledge altogether.

The paper is not being split (or folded for that matter), it's being measured.

I don't recall anyone, in any equation, mentioning matter or energy, and the notion that "energy is always lost" is simply false.

   

Pythagoras argued that all events can be expressed mathematically but much of the mathematics was inaccurate, it was the work of later philosophers mathematicians and physicists to create many of our mathematic expressions suggesting it is possible to express all events mathmatically however that does not mean it can be done.

   

As a matter of fact, Godel's Incompleteness Theorem, provided that no logical system can show all truths to be true or all falsehoods to be false. In other wrods, using mathematics, some truths will remain unknowable. Turing expanded on this concept wiht his Halting Problem. Roger Penrose presented a fascinating argument that only a non-raitonal intelligence could have developed Godel's Incompleteness Theorem (that is to say, a computer could not, even in princinple, discover Godel's Incompleteness Theorem).

Also interesting, in this vain, is the case of the Catholic Church against Galileo. It is a little known fact that the church had virtually resinged themselves to a heliocentric paradigm by the time Galileo came around. That wasn't the main thing the church took Galileo to task for. Galileo alos conjectured that science could know the universe absolutely, and that if the state of all parts were known at a given instant in time, then the future universe could be completely predicted (the "clockwork" universe). The Church had a much bigger problem wiht this statement, since where was the room for God in a detmerministic universe merely running its inevitable course? Of course, Godel (and Heisenberg, and Shrodinger adn chaos theory and others) have shown that the Chuch was right--the universe, such as we know it now, is not deterministic.

   

You alreday posted that.

The infinite series described in this thread is completely accurate - the fact that an inifinite number of paper measurements is impossible has been mentioned - what then, is your issue with it?

Why are you trying to argue against the math when you clearly didn't understand it in the first place?

   

Previous  1  2  3  Next