finite or infinate?

Question

After dropping a book (a math book, for instance) it falls half way to the floor. It will, subsequently, fall three quarters of the way to the floor, then seven eighths of the way, then fifteen sixteenths of the way and so on and so on. I dropped a book about 20 times, and without exception, it went all the way to the floor every time. How is this possible?

why drop it so many times? just to irritate my theology prof, I guess (warped sense of humor). Most appreciative to all. Some great points, lots to think on.

well I still think the final answer lies in the realm of metaphysics, the infinite defining the finite. Thanks again, all.

Answer 1

Everyone missed the point ... just like trying to shut a sliding door: the door has to go halfway, before that a quarter etc - how can a door pass through an infinite number of points or divisions in a finite time. It would seem the door would never move.
The answer is that space and time have a granularity below which you cannot go. There exists a 'smallest' amount of space and time (plank lengths).
Matter 'moves' from one place to another in a myriad tiny (quanta) jumps - not smoothly at all.

Answer 2

Xeno's paradoxes include

Achilles and the tortoise
"You can never catch up."

"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." (Aristotle Physics VI:9, 239b15)

In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet.

If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead.

Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Xeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.

Solution to Xeno's Paradox
Xeno’s Paradox purports to show that in a chase, the chaser can never catch the person or object being chased. In the paradox the Greek hero Achilles graciously gives his slower rival, which legend has turned into ‘The Tortoise’, a head start.

At the start of the race, Achilles is at position 0 while the tortoise is at the 1 kilometre mark. For convenience, imagine that the tortoise can only run half as fast as Achilles.

Common sense may lead you to think that Achilles would overtake the tortoise after running 2 kilometres. However, by the time Achilles reaches the tortoise’s starting point at the 1 km mark, the tortoise has travelled ½ km ahead, which is 1 + ½ km from Achilles’ starting point at position 0. When Achilles reaches the 1½ km point, the tortoise has reached 1 + ½ + ¼ km, and so on.

When, after N steps, Achilles reaches a distance 2 – (½ )N-1 from the start the tortoise is still in the lead because it is at a distance 2 – (½ )N+1 from the start. No matter how big N (the number of divisions of the journey) becomes, Achilles never overtakes the tortoise!

The Solution

The solution involves the observation that the pursuer and the pursued are finite entities.

Say Achilles has a stride length of 2 metres, and the tortoise has a stride length of 1 metre – this tortoise likes to bound along rather than waddle. For convenience, let’s give the tortoise a 10 metre head start. All else being the same (i.e. Achilles and the tortoise takes the same time to make each stride), after 5 strides Achilles is at the 1 km mark, and the tortoise is at the 1.5 km mark. After another 5 strides Achilles will catch up to the tortoise at the 2 km mark.

The crucial thing here is that both Achilles and the tortoise have a finite stride length that remains constant throughout the race. In Xeno’s footrace on the other hand, Achilles and the tortoise make smaller and smaller strides, so that near the 2 km mark they are making infinitessimally small strides. In fact, Achilles never catches up to the tortoise in the Xeno’s Paradox because neither of them ever reaches the 2 km mark.

The upshot of comparing these two descriptions of the race is that for Xeno’s Second Paradox to be true, both the pursuer and the pursued must be able to reduce their ‘stride length’ an infinite number of times.

Finite objects such as people and tortoises can reduce their stride length, but not any smaller than the size of their bodies. So the number of reductions they can achieve is finite.

Assuming that there can be no object of a size between a point and an object with finite size, what this means is that when a finite object moves, it does so in finite increments, and there is a minimum finite amount of movement for each finite object.

Stimulus: "The Infinite Book", by John D. Barrow.

Answer 3

That's an example of a convergent series.
You can add up an infinite number of terms to get a finite number. It's been proved in Calculus; there's no doubt in anyone's mind that it's true. Kind of a weird concept though.

Anyway the proof goes something like this.
Let S be an infinite series such that
S = 1/2 + 1/4 + 1/8 + 1/16 + ...

Double both sides of the above equation so that
2*S = 2*1/2 + 2*1/4 + 2*1/8 + 2*1/16 + ...
= 1 + 1/2 + 1/4 + 1/ 8 +...

Then subtract the first series from the second. Most of the terms fall out because they match exactly.

2*S - S = 1 + 1/2 + 1/4 + 1/ 8 +...
- 1/2 - 1/4 - 1/ 8 +...
S = 1.

So the book falls all the way to the floor.

Answer 4

According to quantum physics, the book never really reaches the floor. The atoms in the floor and book develop repelling charges before they collide releasing energy in the form of heat, sound, light, ect. So, according to quantum physics, your book and you never touch the floor or eachother. In fact, nothing touches anything else, the atoms are just forced to repel the charges of other objects, even under pressure. Don't forget, there's a very small chance that the book will fall through the floor and the world due the correct lining up of the atoms at exactly the right moment. Trippy stuff.

Answer 5

This is an example of a calculus (math) concept called 'limits'.

Another example: sitting on a couch with your friend, and you scoot halfway to them, then halfway again, and so on. Will you ever get to them?

Although you could do this an infinite number of times, the limit you would reach (the other person) is very real and finite, and can be calculated.

The increasingly small factions you mention is an example of an infinite set. It turns out some infinities are larger than others! Go figure.

Answer 6

This is an example of an ancient paradox attributed to Zeno. The thing you need to realize is that the *time* between the different book positions acts in a way similar to the distances. If you do an example with uniform motion (as opposed to accelerated motion), it takes half the time to go half way, three fourths of the time to go three fourths of the way etc. So both distance and time are being divided in a similar way. This allows a finite distance (with an infinite number of points) to be traversed in a finite time (withan infinite number of instances).

Answer 7

this is called Zeno's paradox. However, the basic properties of calculus dictate that the book will reach the floor. Like so where the limit is the distance left to the floor and x is the amount of "falls":

lim 2 ^(-x) = 0
x --> inf.

See, eventurally the distance becomes infintesimal until it reaches zero. Even though there are an infinate amount of "falls," the amount of time spent in "falls" gets infinately short because there is less distance to travel.

Answer 8

I am guessing you are talking about dividing numbers in half, but never reaching zero. True in math, but not true in physics. The best analogy would be a black hole, the material that is in the center keeps compressing because of the gravity it is creating. Eventually it reaches what is called a singularity, where the gravity created by the material still exists, but the material does not. Almost like a cartoon being sucked into a vacuum cleaner.

Answer 9

It isn't true that the distance of the book above the floor is halved as it falls. The distance merely reduces till zero.

A better example where a value is always halved would be the half-life of a radioactive substance. It will never become "non-radioactive" as the number of clicks is always halved over a fixed period of time.

Answer 10

I'm not quite sure if you have posed your problem correctly or perhaps i can't understand it correctly. anyway, without concerning myself with the first part of your question, the book falls all the way to the floor because of gravity. if you did this on the moon, the book would not behave in the same way.