Pick any number A.Hello again,
Well you are saying a lot but not really explaining what you mean.
Maybe give a couple examples?
For A there exists another number B such that:
B >> A.
Pick any number A.Hello again,
Well you are saying a lot but not really explaining what you mean.
Maybe give a couple examples?
Hi,What the double caret means? Never seen that before. Exponent?
Hi,Pick any number A.
For A there exists another number B such that:
B >> A.
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.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).
Works for what???3.14 works for me...
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.
People who think about this sort of thing have come up with some upper bounds: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?
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.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).
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.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.
This sounds like: "Prove all numbers exists by enumerating them."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.
It seems like a largely nonsensical question.
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?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.
You can agree with whomever you wish. It doesnt make it true or false
Says you! I'm an expert cosmetologist.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.
Hi,Says you! I'm an expert cosmetologist.
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.Hello again,
Well you are saying a lot but not really explaining what you mean.
Maybe give a couple examples?
Hi,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.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.
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