Big numbers

MrAl

Joined Jun 17, 2014
11,389
What the double caret means? Never seen that before. Exponent?
Hi,

In some texts it just means exponent, but in others it means iterated exponent so:
2^^3 would mean (2^3)^3
but i dont think i meant it that way just normal exponent. Maybe i typed a second carat by accident.

But you know with all the math symbols out there these days i might have really meant:
34^#(%*9.8_-?kg===<=>pi
:)
 
Last edited:

MrAl

Joined Jun 17, 2014
11,389
Pick any number A.

For A there exists another number B such that:

B >> A.
Hi,

Thanks, but that is YOUR explanation i wanted HIS explanation because we dont know what HE meant yet. It could be the same as yours or different.

Something i like to bring up to think about is what is the largest number that can ever be counted too assume you could count any way you want to even if using a computer.
To start, one guy might start counting at age 5 and when he gets old he passes it to his son, and he gets old and passes it to his son, etc. At some point the universe ends so that last guy got only so high. Then we can reason that if we use a computer, we can count faster and maybe even by 10's. But then why not by 100's, or 1000's. At the end of the universe we'd be higher in the count. But then what other mod's would help. Maybe instead of counting we could multiply integers. Start with 10x10 or maybe 100x100 but then why stop there.
So the question is, given any method we can possibly come up with, at the end of the universe what is the highest number we could ever calculate NUMERICALLY (or if you prefer to stay with counting).
 
Last edited:

WBahn

Joined Mar 31, 2012
29,979
So the question is, given any method we can possibly come up with, at the end of the universe what is the highest number we could ever calculate NUMERICALLY (or if you prefer to stay with counting).
It seems like a largely nonsensical question. Whatever number you come up with, someone else could START their counting by adding twice that number in each iteration.

More interesting might be to consider what the highest number is that we could ever REPRESENT. There are about 10^80 electrons/protons/neutrons in the known universe. How many bits of information could be conceivable represent with each one?

But even this is largely meaningless since what do we mean by "represent"? 10^6745809237642162685 is a number that I just represented and it far, far exceeds the number of particles in the universe and any conceivable number of states those particles could be in. So I just define by representation to mean every larger numbers. For instance, I use the electrons to represent an integer A, the protons to represent B, the neutrons to represent C, and then define my number to be ((10^A)^B)^C. Then I turn around and say that its 10^(A^(B^C)).

So we need to more carefully define what a "number" is or a "representation" is so that we have a chance at having a meaningful question. The simplest might be something like what is the largest integer that we can represent exactly AND that we can represent that integer less one (I think this might be the same as saying that we can represent every integer from zero through that value, but I'm not sure).
 

WBahn

Joined Mar 31, 2012
29,979
3.14 works for me...
Works for what???

How is 3.14 consequentially different than, say, 2.84?

Depending on which aspect of the thread you are responding to, do you mean that 3.14 is a close-enough approximation to π for you, or that it is somehow close enough to be a "big" number for you?
 

WBahn

Joined Mar 31, 2012
29,979
I would agree that, for the vast majority of practical applications, 3.14 is a sufficiently close approximation to π. After all, the error is only about 0.05%, or less than one part in a thousand. Relatively few applications require accuracies better than 1%. But, certainly, such applications DO exist.
 

bogosort

Joined Sep 24, 2011
696
More interesting might be to consider what the highest number is that we could ever REPRESENT. There are about 10^80 electrons/protons/neutrons in the known universe. How many bits of information could be conceivable represent with each one?
People who think about this sort of thing have come up with some upper bounds:

Information packing: \( 10^{69} \) bits per square-meter (assuming we could store a bit per Planck length).
Information rate: \( 10^{44} \) operations per second.

Using these limits, Scott Aaronson calculated a maximum of \( 10^{122} \) bits allowed in any physically-realizable computation. If a computation required more bits, some of the bits would end up receding from us faster than the speed of light.

So we need to more carefully define what a "number" is or a "representation" is so that we have a chance at having a meaningful question. The simplest might be something like what is the largest integer that we can represent exactly AND that we can represent that integer less one (I think this might be the same as saying that we can represent every integer from zero through that value, but I'm not sure).
Indeed, and I believe your intuition is correct. In general, a number \( k \) will require \( \log_2{(k)} + 1 \) bits to encode. However, some numbers have a representation that is compact (in the sense of Kolmogorov complexity) in a given base and require only a handful of bits to encode. We may think of this as the mostly-zeros class of numbers, which -- by virtue of an intrinsic repitition in the chosen base -- is symbolically compressible. For example, the number \( 10^{1000} \) has a representation that can be encoded in 3,300 bits. But, as the last thousand of its 1,001 digits are zero, we can easily compress its encoding to a few bits. On the other hand, if we randomly set each of those thousand digits to a value between 0 and 9, we will almost certainly end up with a number that cannot be symbolically compressed: the shortest representation of the number is the number itself.

And though the mostly-zeros class of numbers are easy to label, we nonetheless need their full representation in order to use them in any computations. For example, I can "invoke" the number \( 2^{(10^6)} \) with a few bits-worth of symbols. I can even mentally construct it, say, by thinking of the cardinality of the powerset of a set of a million things, and this equivalent representation requires only a few more bits of information. But to actually compute with this number would require its full representation in a million bits.

I think the intuition is that some number \( k \), being the successor of another number \( k - 1 \), is in some sense comprised of all the numbers before it, and so carries each of them with it. Put another way, the "state-space" of \( k \) includes \( k - 1 \) states, one for each predecessor number. And so, while we as humans can label or even construct numbers that far exceed the informational capacity of the universe, we can't actually use them outside the context of labels.
 

MrAl

Joined Jun 17, 2014
11,389
It seems like a largely nonsensical question. Whatever number you come up with, someone else could START their counting by adding twice that number in each iteration.
But see my text covers all those scenarios. Read it again and you'll see that it's not just about counting and numbers (assume integers for now) it's also about time.
In the case you brought up i already suggested as a possibility but it does not matter because at some point the universe ends and that's all time we get.

Now as a secondary thought, imagine that your second someone started by adding twice the number in each iteration but then someone else a 3rd party realizes that they can add three times the number in each iteration, but see again that discovery in itself took time. So the next guy the 4th party with his/her idea adds more time to the whole process and the 5th guy etc., but at some point the Nth guy dies at the end of the universe and that's that. We've reached the highest number.

Now consider that what we both have written even took time. If we keep writing back and forth better and better ideas for the counting process, we've used up time also so we are still limited due to the end of the universe. Even if we use very large numbers to start with.

Now consider even that we may not be the only universe and scientists not involved in the counting process discover another universe. If we or the last guy can get to that universe maybe he can continue the process, until that universe ends, or maybe the last person in that universe (that took over the task) can get to another one also, etc.

So really this isnt about counting so much as it's about trying to figure out what would be the fastest process to use and what natural limits might be imposed on the highest possible number.

In the end we keep coming up with better and better ideas but at the same time more time has passed so we are also constantly getting closer to the natural limit, if in fact there is one.
 

joeyd999

Joined Jun 6, 2011
5,237
But see my text covers all those scenarios. Read it again and you'll see that it's not just about counting and numbers (assume integers for now) it's also about time.
In the case you brought up i already suggested as a possibility but it does not matter because at some point the universe ends and that's all time we get.

Now as a secondary thought, imagine that your second someone started by adding twice the number in each iteration but then someone else a 3rd party realizes that they can add three times the number in each iteration, but see again that discovery in itself took time. So the next guy the 4th party with his/her idea adds more time to the whole process and the 5th guy etc., but at some point the Nth guy dies at the end of the universe and that's that. We've reached the highest number.

Now consider that what we both have written even took time. If we keep writing back and forth better and better ideas for the counting process, we've used up time also so we are still limited due to the end of the universe. Even if we use very large numbers to start with.

Now consider even that we may not be the only universe and scientists not involved in the counting process discover another universe. If we or the last guy can get to that universe maybe he can continue the process, until that universe ends, or maybe the last person in that universe (that took over the task) can get to another one also, etc.

So really this isnt about counting so much as it's about trying to figure out what would be the fastest process to use and what natural limits might be imposed on the highest possible number.

In the end we keep coming up with better and better ideas but at the same time more time has passed so we are also constantly getting closer to the natural limit, if in fact there is one.
This sounds like: "Prove all numbers exists by enumerating them."

I agree with @WBahn when he said:

It seems like a largely nonsensical question.
 

bogosort

Joined Sep 24, 2011
696
And so, while we as humans can label or even construct numbers that far exceed the informational capacity of the universe, we can't actually use them outside the context of labels.
Not to have a conversation with myself, but this brings up an interesting question. Presumably, there is some number \( n \) that is physically-realizable as a set of bits, yet the \( 2^n \) intermediate states of an \( n \)-qubit quantum computer exceed the computational power of the universe. So, even though (per the Holevo bound) we never access more than \( n \)-bits of information -- and so the computation is physically possible -- where exactly does the universe store those intermediate states?
 

MrAl

Joined Jun 17, 2014
11,389
This sounds like: "Prove all numbers exists by enumerating them."

I agree with @WBahn when he said:
You can agree with whomever you wish. It doesnt make it true or false :)
You have to be able to appreciate these kinds of questions about reality. If you think that is nonsensical, then you havent been paying much attention to modern physics and the study of the cosmos.
 

joeyd999

Joined Jun 6, 2011
5,237
You can agree with whomever you wish. It doesnt make it true or false :)
You have to be able to appreciate these kinds of questions about reality. If you think that is nonsensical, then you havent been paying much attention to modern physics and the study of the cosmos.
Says you! I'm an expert cosmetologist.
 

MrAl

Joined Jun 17, 2014
11,389
Says you! I'm an expert cosmetologist.
Hi,

Ha ha, good one ! :)

Seriously though when i bring that up it usually provokes a big discussion usually interesting where everybody tries to come up with better and better ideas to get as high as possible as fast as possible.
Computer counting followed by a transfer to a newer computer with faster processor, etc.
Maybe i didnt make the point of the discussion correctly then appologies that's my bad :)
 

DarthVolta

Joined Jan 27, 2015
521
Hello again,

Well you are saying a lot but not really explaining what you mean.
Maybe give a couple examples?
I just mean any number of finite size, could be called small. And visually I just picture zooming out on a number line 'forever'. So all numbers might as well be called small. I guess that's what finite size means though.
 

MrAl

Joined Jun 17, 2014
11,389
I just mean any number of finite size, could be called small. And visually I just picture zooming out on a number line 'forever'. So all numbers might as well be called small. I guess that's what finite size means though.
Hi,

Well if you ask me that view sounds a little too general. I think it depends highly on the ratio of the two numbers under consideration. And in fact that is how limits are evaluated that involve division. Of course all finite numbers are less than infinity, but a finite 1000 is not that much less than the finite 1001. So the ratios 1000/1001 or 1001/1000 are both close to 1 while inf/1001 is infinity and 1001/inf is zero.
 

WBahn

Joined Mar 31, 2012
29,979
I just mean any number of finite size, could be called small. And visually I just picture zooming out on a number line 'forever'. So all numbers might as well be called small. I guess that's what finite size means though.
I think making that claim robs the term "small" of any meaning. As with most things, context is important. So WHEN you are talking about comparing infinity to a finite number, it may well make sense to talk about any finite number being small IN COMPARISON. But that is not the only context, so I don't see it as being useful to think of any finite number as being categorically small.
 
Top