Factorial (n) for n > 24 is greater than 10^n.
24! is approximately Avogadro's number (about a 3% difference).
How big are factorials?
21–30 of 43 posts
Re: How big are factorials?
#22My favorite one is with the 52! seconds:
Start a timer that will count down the number of seconds from 52! to 0. Then walk around the Earth’s equator with one step every billion years. Then, after you make your way around the earth equator (by taking 1 step every billion of years), you take one drop of water out of the Pacific Ocean. Then, you repeat the process of walking around the equator, and everytime you walk around, you keep draining one singular drop of water. After the ocean is fully drained, you refill the ocean and put a piece of paper underneath you. Now, you once again repeat this process of walking, draining, and placing papers. After your stack of papers has reached the Sun, you repeat another 1000 times.
After all this, you have completed just about a third of the timer.
Re: How big are factorials?
#23What's surprising (to many) is that n! < exp(n log n)
> What's surprising (to many) is that
> n! Why would that be surprising? I can see that many wouldn’t know whether it’s true, but
exp(n × log n) =
exp(log(n) × n) =
exp(log(n))^n =
n^n
1 × 2 × 3 × 4 × … × n
Re: How big are factorials?
#24What's surprising (to many) is that n! < exp(n log n)
[deleted]
Re: How big are factorials?
#25lg(n!) grows roughly as (n lg n). Constants matter, of course, but to that's the rough estimate. As an aside, if you take numbers from 0 to (n-1) in an array, there are n! configurations, so representing each configuration or differentiating each configuration take n lg n bits. So, in some sense, taking a mapping that's able to differentiate the input state to map to the ordered state takes at least O(n lg n) time, t…
My kids love taking about big numbers. TREE(3) is a family favorite. So, I was going over sequences with them, and I decided to go slow instead. My sequence was: 1 1 1 1 ... 1 ...
They accused me of using just "all 1s" (which is, naturally, cheating). Ai contraire!
The count of the number of digits in the decimal representation of the number of unique primes in the prime factorization of the natural numbers.
The best part is that even pretty young kids can compute this sequence; by the first "2" is at 2*3*5*7*11*13*17*19*23*29!
(Hopefully I got that right; the phone doesn't make it easy to type!)
Re: How big are factorials?
#26What's surprising (to many) is that n! < exp(n log n)
> What's surprising (to many) is that > n! Why would that be surprising? I can see that many wouldn’t know whether it’s true, but exp(n × log n) = exp(log(n) × n) = exp(log(n))^n = n^n and it’s not surprising that 1 × 2 × 3 × 4 × … × n for n > 1
I don't know why that is surprising, but I'm teaching computer science at a university, and students seem surprised every time it comes up.
Re: How big are factorials?
#27My favorite one is with the 52! seconds: Start a timer that will count down the number of seconds from 52! to 0. Then walk around the Earth’s equator with one step every billion years. Then, after you make your way around the earth equator (by taking 1 step every billion of years), you take one drop of water out of the Pacific Ocean. Then, you repeat the process of walking around the equator, and everytime you walk a…
For another perspective, 52! is roughly the number of atoms in a galaxy. Galaxies are really quite large!
Re: How big are factorials?
#28My favorite one is with the 52! seconds: Start a timer that will count down the number of seconds from 52! to 0. Then walk around the Earth’s equator with one step every billion years. Then, after you make your way around the earth equator (by taking 1 step every billion of years), you take one drop of water out of the Pacific Ocean. Then, you repeat the process of walking around the equator, and everytime you walk a…
Holy hell, this is great.
I once made a little tool for getting more intuitive spatial scales for things in the universe at https://observablehq.com/@ikesau/scale-to-the-universe
I feel like you could do something similar for these sorts of "fathom this large number" recipes.
Re: How big are factorials?
#29The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years ago https://czep.net/weblog/52cards.html Anyone know how to determine the age of this page (it's got be at least 20yrs old)
To me it feels like a smuggled exponentation. If I take half of 52 and raise it to itself and add a base unit to it like grams or meters, that's an incomprehensibly large amount.
Re: How big are factorials?
#30lg(n!) grows roughly as (n lg n). Constants matter, of course, but to that's the rough estimate. As an aside, if you take numbers from 0 to (n-1) in an array, there are n! configurations, so representing each configuration or differentiating each configuration take n lg n bits. So, in some sense, taking a mapping that's able to differentiate the input state to map to the ordered state takes at least O(n lg n) time, t…
My kids love taking about big numbers. TREE(3) is a family favorite. So, I was going over sequences with them, and I decided to go slow instead. My sequence was: 1 1 1 1 ... 1 ... They accused me of using just "all 1s" (which is, naturally, cheating). Ai contraire! The count of the number of digits in the decimal representation of the number of unique primes in the prime factorization of the natural numbers. The best…
I've struggled to understand how TREE(3) works and why it grows so big. I once posted to Math.Stackexchange many years ago and got a professor to help explain it to me: https://math.stackexchange.com/questions/313134/why-is-tree3...
I also loved David Metzler's series on ridiculously big numbers: https://www.youtube.com/playlist?list=PL3A50BB9C34AB36B3