Finitum and infinitum

[cronopio]

d=2*pi*r^2 where d=circle diameter and r=circle radius

How can I get a finite number by multiplying two finite numbers by an infinete number? I try to write it down in a paper and I do not get it.

One more question related with this infinitum-finitum matter...While I am writing this message I have one mirror in front of me and one mirror at my back.

I am seeing infinite (I have not counted them but I suppose so) front sides and back sides of me. If I move my hand, and infinite number of hands move, and infinete number of fotoms are moving with my hand.

Fotons=energy, infinety fotons=infinety energy but the energy of the universe, as far as I know, is finite, show there is not such thing as infinite Miguels watching each other at a mirror.

Where am I getting wrong? Please answer with a short, finite answer.

Kind Regards,
Miguel

[Pythagoras]

A partial answer:

I think your definition of finite and infinite is flawed. Pi is transcendental, it can not be expressed as a fraction. There is no thing as an 'infinite' number. Infinity is more of a concept.

Understanding Zeno's paradox might help you grasp the difference between things that are finite and infinite. Basicly the paradox states a runner must run half the distance between two points, then half the remaining distance, and so on. He must continually half the distance remaining forever, hence a runner can not get to his destination. Since we all must do this to get between two points, movement is impossible it would seem! Not really. In mathematics we learn that adding 1/2+1/4+1/8 + 1/16 (continually halving the distances) equals one. Also, when moving from two points, the time it takes to cover half distances is not constant, so it wont take forever to move.

[Rysto]

Pythagoras: In mathematics we learn that adding 1/2+1/4+1/8 + 1/16 (continually halving the distances) equals one.

Excuse me? No matter how many times you halve the distances and add them to your total, you will never reach 1. Unless you round.

[Rysto]

Oops, forgot about the OP. Like Pythagoras said, Pi is not an "infinite number". Pi is an irrational number, a number that cannot be expressed as a ratio, ie a fraction. Consequently they have an infinite number of decimal places, which is probably where your confusion lies. In any case, when you multiply Pi by 2 and r^2, you do end up with another irrational number.

[Pythagoras]

Originally posted by Rysto
Pythagoras: In mathematics we learn that adding 1/2+1/4+1/8 + 1/16 (continually halving the distances) equals one.

Excuse me? No matter how many times you halve the distances and add them to your total, you will never reach 1. Unless you round.

Let me rephrase it then. And its actually supposed to be 2, my bad.

The sum:
1/2+1/4+1/8+1/16+....+1/2^n
converges to 2 as n approaches infinity.

[stankow]

I think you were right the first time, Pythagoras. Unless I'm missing something, that sum converges to 1.

[ultrafilter]

stankow's right.

[bibliophage]

Originally posted by cronopio
d=2*pi*r^2 where d=circle diameter and r=circle radiusI think you're a little mixed up there.

d = 2r
A = pr2

[The Great Unwashed]

Originally posted by Rysto
[BIn any case, when you multiply Pi by 2 and r^2, you do end up with another irrational number. [/B]
Not necessarily: what if you choose r to be 1/sqrt(Pi)

Then r^2 = 1/Pi
Pi * r^2 = Pi/Pi = 1
2 * Pi * r^2 = 2

So maybe this answers part of the OP (or at least I hope it does)

How can multiplying a number with an infinite decimal expansion (Pi) with other numbers give a finite decimal expanded answer.

[Pythagoras]

Well the numbers are concepts independant of how they are represented. It just so happens that pi can not be exactly represented in a decimal format. It can, however, be represented as an infinite sum exactly.

Hmm I thought it was 1, some site said it was 2. Well it converges to a constant, thats whats important. Moreover each step in the addition does not correspond to a constant time, if it did (if you did each half in 1 second) you indeed would never approach the other side, but since you spend decreasing time on each half then motion is possible.

[DrMatrix]

Originally posted by Pythagoras

There is no thing as an 'infinite' number. Infinity is more of a concept. Well, that depends on how you define number. There are transfinite ordinal and cardinal numbers. À0 being the smallest transfinite cardinal number.

The numbers in the OP are finite though.

[Nametag]

pythagoras, that series is usually expressed with 1 as the first term in which case it converges to 2.

Anyway, Bibliophage caught the same thing that caught my eye: d=2r.

[cronopio]

The Great Unwashed has got the point that surprise me: The close relation between finitum and infinitum when I thought they were completely separate concepts.

As he said,

"How can multiplying a number with an infinite decimal expansion (Pi) with other numbers give a finite decimal expanded answer"

The other example I put goes into the same direction: From two mirrors and one apple I can get an infinite number of images of that apple.

Beautifully, in this case 3 equals infinitum.

A silly game that got into my mind after several drinks that night.

Nametag, Bibliophage you are absolutely right d=2r. I´ve been pretty worried about this lapsus (should I go to primary school again?), but hey, if we are in a world where 3 can be infinitum, everything is possible, even d=2*pi*r^2

[ultrafilter]

Originally posted by cronopio
The other example I put goes into the same direction: From two mirrors and one apple I can get an infinite number of images of that apple.

Beautifully, in this case 3 equals infinitum.

A silly game that got into my mind after several drinks that night.

Now let's talk about the sort of things that get in your mind after a few drinks. You take an apple and two mirrors, and no matter how you arrange them, you've still got an apple and two mirrors. There is no "infinitum"*, there's just three objects. You may see more apples than you can count, but that doesn't mean that they exist, or even that you're seeing an infinite number.

As to the other end of the OP, the difference between p and, say, 3, is that 3 has a finite-length decimal expansion, and p doesn't. Since both of them have an absolute value less than 4, they're both finite.

* And I've never heard the word "infinitum" before, not even when studying transfinite numbers. Can you supply a definition, or is it your own term?

[Chronos]

Heck, why use something complicated like p? One third also has an infinite decimal expansion, but 0.3333... * 3 = 1 . The fact that one third doesn't have a finite decimal expansion doesn't really tell us anything about the number one third, just about our way of expressing it.

[Rysto]

Infinitum is latin for infinite, I think.

[Animal]

Infinity is that which is not finite.