Comment by fwip
8 months ago
At very large numbers, even ratios don't really matter.
For instance, if you personally owed $100 trillion, you wouldn't be much relieved by a court order that reduced your liability by 99%. Or, if you're looking at numbers in scientific notation, you don't much care about the difference between 2e40 and 5e40.
In this case, the ratio is around 10^200. An incomprehensibly vast number, to be sure.
But because tetration is the next operator up from exponentiation (the way exponents are from multiplication), any fixed divisor ceases to "matter" very quickly. The difference between 10^^10,000,000 and 10^^10,000,001 is (10^^10,000,000 to the tenth power), if my understanding is right.
There's basically no way to get it into comprehensible territory even with repeated divisions. 10^^1 = 10, 10^^2 = 10^10 (ten billion), and 10^^3 is 10^(10^10) = 10^10,000,000. Already, dividing by 10^200 isn't going to meaningfully affect your number (10^99,999,800).
10^^10,000,000 is that kind of incomprehensible growth that we just saw from 1 to 2 to 3, repeated 10 million times.
> For instance, if you personally owed $100 trillion, you wouldn't be much relieved by a court order that reduced your liability by 99%.
It is *never• true that differences don’t matter. Only true that in some respects the difference matters, others it does not.
You manufactured a reasonable situation for differences not mattering.
But if I had $1 trillion, 99% off $100 trillion would matter.
As I noted, from the perspective of anyone in those universes, a 1 grain universe, or a solid grain universe would each be a spotty context to make a living.
But in very different ways!
So in this case, the ratio between two incomprehensibly large numbers, happens to be highly comprehensible under the circumstances in which they were described. I.e. universes and grains.
One can imagine that one of unexplained constants of nature might be a result of differences between unimaginably large numbers. Which again shows, that there is no such things as numbers so large differences don’t matter. Only cases where they don’t matter, or do. As with all approximations.