# Thread: The legitimacy of randomizing infinity: Rolling a sphere dice.

1. When i random infinity in a calculator, it gives me an error.
same for C++ rand() function

Why not?

Equipment:
1) If i have a dice with infinite sides, 1 through infinity so, a sphere.
2) The size of the side of the die is
arbitrarily close to, but greater than: Zero.
3) My die is a perfect sphere, so it has infinity sides.

Technical Reference:
1) The probability of rolling and getting any number is 1. Its because when I roll my die, it must return one value. Thus 1 will he the numerator.
2) The number of numbers we may get is infinity, because my dice has infinity sides. Infinity will me the denominator

Scenario:
1) When i roll a sphere, it stops with a point.
2) in this case, I get the number 2. arbitrarily close to, but greater than Zero.
3) The probability of me getting this number:
1/infinity=infinitesimal.
4) Which share the same probability to roll the number 10*10^100
5) Which share the same probability to roll the number 99*10^999

Randomizing Infinity:
Since randomizing infinity can return a proper value, e.g. 5, isn't randomizing infinity legitimate?

Other questions: Is infinity a valid value? Is infinitesimal a valid value? If no, then why is it possible for me to roll a sphere die? If yes, why is it/not a specific value?

2.

3. Originally Posted by RamenNoodles
When i random infinity in a calculator, it gives me an error.
same for C++ rand() function
How are you entering infinity in a calculator or in C++. As far as I know, neither has the capability to represent infinity.

Since randomizing infinity can return a proper value, e.g. 5, isn't randomizing infinity legitimate?
If the question is, "is it possible to have random numbers between 0 and infinity?" then the answer is yes.

If the question is "is it possible to have random numbers between 0 and infinity in a computer?" then the answer is no. Because a computer can only represent a finite number of values.

Other questions: Is infinity a valid value?
No. It is larger than any numerical value. (Note that there is more than one infinity; each infinity is infinitely larger than the previous.)

Is infinitesimal a valid value?
I assume not: it would be smaller than any real number.

4. Originally Posted by Strange
Originally Posted by RamenNoodles
When i random infinity in a calculator, it gives me an error.
same for C++ rand() function
How are you entering infinity in a calculator or in C++. As far as I know, neither has the capability to represent infinity.
rand() % 1/0;

5. Originally Posted by RamenNoodles
rand() % 1/0;
1/0 is mathematically undefined (not infinity). And therefore generates an error in a computer or calculator. Nothing to do with rand().

6. No. It is larger than any numerical value. (Note that there is more than one infinity; each infinity is infinitely larger than the previous.)
haha i read that up years ago, about countable infinity, etc

and listing all the decimal places and then changing 1 of each from every other number, makeing a whole new number.

7. if 1/0 infinity,
and 1/infinitesimal = infinity,
infinitesimal ≠ 0.

i guess that's where the problem is when people use the: "The result gets bigger when you keep decreasing the denominator. Keep decreasing it until it becomes zero."

If a side in my die has zero area, infinite sides of my dice will not produce an area.
Since the infinite sides of my die must have an area since my die is a sphere and has an area, each side is infinitesimally small.

I found out the problem, that I should not divide by zero, but infinitesimal, as the surface area isn't zero, but infinitesimal.

8. Technically, is any supercomputer in the distant future ever able to represent infinity?
Its so biggggg

9. No

10. Originally Posted by RamenNoodles
Technically, is any supercomputer in the distant future ever able to represent infinity?
Its so biggggg
I don't see why it couldn't be done today. The modern theory of transfinite cardinal and ordinal numbers is well established in mathematics. It's done using finite strings of symbols that obey the rules of logic and the axioms of set theory.

If a mathematician can be trained to compute Aleph-null + Aleph-null = Aleph-null, a computer could easily be programmed to do the same. In fact it would be a fairly easy elementary programming exercise to write a computer program that implements cardinal or ordinal arithmetic. You could have a checkbox yes/no to control whether or not you want to use the Generalized Continuum Hypothesis to allow the reduction 2^Aleph-n = Aleph-(n+1). I'd be surprised if a grad student isn't already working on this. We have computer systems for symbolic algebra; and cardinal and ordinal arithmetic aren't any more complicated than that.

In fact we already have sophisticated computer systems that do symbolic integration. That's a much harder programming trick than transfinite arithmetic, which is in fact very simple.

So what's going on here? Why is this counterintuitive task suddenly so obviously easy? Well, think about how we represent ordinary numbers in a computer. When we represent the number 5 in a computer, we don't actually have the number 5 itself inside the computer. The number 5 is, after all, only a mental abstraction. It's not physical, hence could never be implemented in hardware.

Nor do we have 5 things in the computer -- 5 bits, 5 registers, whatever -- allocated to representing the number 5.

No.

What we have instead is a symbolic representation of the number 5 in some particular notational system -- usually binary representation with 1- or 2-complement arithmetic; along with some rules, programmed into the computer, that define how to manipulate these representations. So if we have the internal representation for 5 and the internal representation for 3, the computer knows how to manipulate the notation to achieve the result 5 + 3 = 8.

To use a computer to work with infinity, we do not need to have infinity inside the computer; nor do we need infinitely many bits or infinitely many hardware components of any type.

All we need is a finite notational system for infinity, along with some unambiguous, deterministic rules for how to manipulate the notation. And this we've already got, thanks to the pioneering set theorists like Cantor and Zermelo and others, who worked out the rules of transfinite arithmetic more than a century ago.

These rules are very simple. They tell you exactly how to add, multiply, and exponentiate transfinite cardinals and ordinals. It would actually be very straightforward to implement a calculator to perform transfinite arithmetic.

Wikipedia has a stub article that links to more detailed articles on cardinal and ordinal arithmetic.

Transfinite arithmetic - Wikipedia, the free encyclopedia

11. Originally Posted by someguy1
To use a computer to work with infinity, we do not need to have infinity inside the computer; nor do we need infinitely many bits or infinitely many hardware components of any type.

All we need is a finite notational system for infinity, along with some unambiguous, deterministic rules for how to manipulate the notation. And this we've already got, thanks to the pioneering set theorists like Cantor and Zermelo and others, who worked out the rules of transfinite arithmetic more than a century ago.

These rules are very simple. They tell you exactly how to add, multiply, and exponentiate transfinite cardinals and ordinals. It would actually be very straightforward to implement a calculator to perform transfinite arithmetic.

Wikipedia has a stub article that links to more detailed articles on cardinal and ordinal arithmetic.

Transfinite arithmetic - Wikipedia, the free encyclopedia
Transfinite arithmetic - Wikipedia, the free encyclopediawow.. wikipedia contains amazingly little about transfinite arithmetic.

and if a calculator doesnt store infinity, but only store a representation of infinity, how can it manipulate with other numbers, e.g. (inf-10 or inf*20 or inf^3 or inf/0)?
since infinity disobeys some laws of maths in nature e.g.(inf+1=inf)
don't we have to store infinity to perform such calculations?

12. Its legitimate, but you have to be able to define the probability for getting a particular value, and it must be non-zero/non infinitessimal. So for instance, Keep flipping a coin, until you get heads. the number of tails you get will be a number between zero and infinity, but its just unlikely that it will be more than a few. You can write a computer program where the probability of halting is way smaller, and it will be more likely to higher number between 0 and inf. For instance, a coin toss program where the odds of halting (heads) is 1/100 per toss will more likely give you two digit numbers... But any number is "possible". Another way you could do it is make a program that spews out random digits 0-9, and keeps appending them to a number, along with some very small probability the process will halt at some point. That gives you big numbers.

edit: looking at the last example, you can see the problem with giving all numbers an equal probability: it results in an infinitessimally small halting probability. The results is the program spews out digits forever, not very useful for a calculator.

13. Originally Posted by TridentBlue
and it must be non-zero/non infinitessimal.
Doesn't a sphere have infinity sides?

14. Originally Posted by RamenNoodles
Transfinite arithmetic - Wikipedia, the free encyclopediawow.. wikipedia contains amazingly little about transfinite arithmetic.
As I said, it's a stub article. It contains links to two detailed articles, one on cardinal arithmetic and one on ordinal arithmetic.

http://en.wikipedia.org/wiki/Ordinal_arithmetic

http://en.wikipedia.org/wiki/Cardina...nal_arithmetic

The article on ordinal arithmetic is a standalone article. The article on cardinal arithmetic is a subsection in the article on cardinal numbers. So Wikipedia's a bit of a mess in terms of the organization of this material, but the information is there.

It's just symbolic manipulation of symbols. We can teach it to undergrads, we can program it in computers.

Originally Posted by RamenNoodles
and if a calculator doesnt store infinity, but only store a representation of infinity, how can it manipulate with other numbers, e.g. (inf-10 or inf*20 or inf^3 or inf/0)?
since infinity disobeys some laws of maths in nature e.g.(inf+1=inf)
don't we have to store infinity to perform such calculations?

The question that you asked was:

Technically, is any supercomputer in the distant future ever able to represent infinity?

And of course the answer is obviously yes. We can teach a high school student to represent infinity using the sideways 8 notation. We can teach college undergrads in set theory class to do cardinal and ordinal arithmetic. These are algorithmic manipulations of symbols -- exactly the kind of thing that computers are very good at.

So the rule is: If a number is finite, you use finite arithmetic. If a number is transfinite, you use the rules of transfinite arithmetic. No problem at all for a computer.

Of course a computer could not "store infinity," any more than it could store the number 5. The only thing a digital computer can ever do is store symbolic representations of things; not the things themselves.

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