You have exposed the fatal flaw in searching for big numbers. What we should really be looking for are big numbers that have a mathematical importance. This, of course, rules out a googol, googolplex or any number derived from that source.
So, what is the biggest number of mathematical importance? In my post viewtopic.php?p=725947#p725947 it has to be the doppelgängion, i.e. 10 to the power of 10 to the power of 68. 10^10^68 is 1 followed by 100 million trillion trillion trillion trillion trillion zeroes.
I did come across some other big numbers while researching for this thread. I'll have to see if I can find them again.
As I get older I think a lot about the hereafter - I go into a room and then wonder what I'm here after.
Inside every old person is a young person wondering what the hell happened.
"Trying is the first step towards failure" ~ Homer J Simpson
OK, here we go. It's actually beyond my comprehension so I'm just going to shamelessly pinch the explanations wholesale.
Graham's Number
All these numbers pale in comparison to Graham's number, a number so large that simply trying to remember all the digits would turn your head into a black hole. The number, which at one point was the largest number to ever be used in a math proof, arose in response to a simple riddle about how to allocate people to a certain set of committees with a few constraints.
While mathematicians are confident that at least 13 people are needed to solve the problem, in the 1970s mathematician and juggler Ronald Graham deduced that the number of people had to be lower than Graham's number. Simply computing the number would take 64 steps, and involves multiplying an insanely huge number of 3s.
There is no way to write the number out using scientific notation, and instead it must be written with a series of up-arrows that denote towers of exponents. Later on, Graham showed that the upper bound to for this riddle is much much smaller than Graham's number, but still huge.
TREE(3)
While Graham's number was one of the largest numbers proposed for a specific math proof, mathematicians have gone even bigger since then. In 1998 the logician Harvey Friedman of Ohio State University proposed a riddle asking how long a sequence of letters needs to be given certain parameters of repeating stretches of letters. While the answer isn't infinite, it's absolutely massive.
The number Friedman derived, TREE(3), is calculated by creating increasingly massive towers of twos raised to the power of two using something called Ackerman functions. To give a sense of the scale, the fourth Ackerman functions involves raising two to the power of 65,536 twos. But TREE(3) is massively, massively larger than that — so massive that it makes Graham's number look like the tiniest fleck of dust in comparison.
"These higher levels of largeness blur, where one is unable to sense one level of largeness from another," Friedman wrote in his paper.
There's also Skewes's number. Well, actually Skewes's numbers as there are several solutions to the number theory, This from Wkipedia.
n number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which
pi (x)>li (x)
where π is the prime-counting function and li is the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing near
e^727.95133<1.397 x 10^316. It is not known whether it is the smallest.
This is really so far past my understanding that there's a fair chance I've killed this thread stone dead!
As I get older I think a lot about the hereafter - I go into a room and then wonder what I'm here after.
Inside every old person is a young person wondering what the hell happened.
"Trying is the first step towards failure" ~ Homer J Simpson
Paul-R wrote: 14 Aug 2022, 12:19OK, here we go. It's actually beyond my comprehension so I'm just going to shamelessly pinch the explanations wholesale.
<much snippage>
This is really so far past my understanding that there's a fair chance I've killed this thread stone dead!
The trouble with those mathematical doodlings is - it's such a fine line between the surreal and the actual, that they wouldn't look out of place in a Douglas Adams book
Paul-R wrote: 14 Aug 2022, 12:19OK, here we go. It's actually beyond my comprehension so I'm just going to shamelessly pinch the explanations wholesale.
Graham's Number
All these numbers pale in comparison to Graham's number, a number so large that simply trying to remember all the digits would turn your head into a black hole. The number, which at one point was the largest number to ever be used in a math proof, arose in response to a simple riddle about how to allocate people to a certain set of committees with a few constraints.
While mathematicians are confident that at least 13 people are needed to solve the problem, in the 1970s mathematician and juggler Ronald Graham deduced that the number of people had to be lower than Graham's number. Simply computing the number would take 64 steps, and involves multiplying an insanely huge number of 3s.
There is no way to write the number out using scientific notation, and instead it must be written with a series of up-arrows that denote towers of exponents. Later on, Graham showed that the upper bound to for this riddle is much much smaller than Graham's number, but still huge.
TREE(3)
While Graham's number was one of the largest numbers proposed for a specific math proof, mathematicians have gone even bigger since then. In 1998 the logician Harvey Friedman of Ohio State University proposed a riddle asking how long a sequence of letters needs to be given certain parameters of repeating stretches of letters. While the answer isn't infinite, it's absolutely massive.
The number Friedman derived, TREE(3), is calculated by creating increasingly massive towers of twos raised to the power of two using something called Ackerman functions. To give a sense of the scale, the fourth Ackerman functions involves raising two to the power of 65,536 twos. But TREE(3) is massively, massively larger than that — so massive that it makes Graham's number look like the tiniest fleck of dust in comparison.
"These higher levels of largeness blur, where one is unable to sense one level of largeness from another," Friedman wrote in his paper.
There's also Skewes's number. Well, actually Skewes's numbers as there are several solutions to the number theory, This from Wkipedia.
n number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which
pi (x)>li (x)
where π is the prime-counting function and li is the logarithmic integral function. Skewes's number is much larger, but it is now known that there is a crossing near
e^727.95133<1.397 x 10^316. It is not known whether it is the smallest.
This is really so far past my understanding that there's a fair chance I've killed this thread stone dead!
Infinity rules!
