Big Numbers.
Jun. 12th, 2006 01:08 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
swestrupThere are, in mathematics, some big numbers. Amongst the biggest (as we're not counting transfinites as numbers here) is Grahams Number. This number is mind-bogglingly, ultra-swattingly big.
Lets put it this way, Knuth invented a notation so you could continue exponentiation. Since a+a+a+a (b times), can be written a*b, and a*a*a*a (b times) can be written, a^b, he introduced a^^b for a^(a^(a^a)) (b times) and a^^^b for a^^a^^a^^a (b times) and so on. This is used for the smaller sorts of big numbers.
Conway introduced his chained numbers which are much, much bigger. In fact the simple a -> b -> n, stands for a | a | a | a (b times) where each | is ^^^^(n times), in Knuth's notation. Adding each extra arrow goes meta recursive on the concept.
Grahams Number is too big (!!!) to be conveniently described by chained numbers, but if we defined f(n) = 3->3->n and using the convention that fk(n) = f(f(f(f(n)))) (k recursive invocations), then Grahams Number is f64(4).
This is believed to be the biggest number that has ever been used in a proof. Graham proved that a certain quantity MUST be less than this number. Considering that the traditional best guess at the actual value Graham was looking for was probably 6, this is also one of the worst upper bounds ever devised (although its recently been shown that the number must actually be over 11).
Lets put it this way, Knuth invented a notation so you could continue exponentiation. Since a+a+a+a (b times), can be written a*b, and a*a*a*a (b times) can be written, a^b, he introduced a^^b for a^(a^(a^a)) (b times) and a^^^b for a^^a^^a^^a (b times) and so on. This is used for the smaller sorts of big numbers.
Conway introduced his chained numbers which are much, much bigger. In fact the simple a -> b -> n, stands for a | a | a | a (b times) where each | is ^^^^(n times), in Knuth's notation. Adding each extra arrow goes meta recursive on the concept.
Grahams Number is too big (!!!) to be conveniently described by chained numbers, but if we defined f(n) = 3->3->n and using the convention that fk(n) = f(f(f(f(n)))) (k recursive invocations), then Grahams Number is f64(4).
This is believed to be the biggest number that has ever been used in a proof. Graham proved that a certain quantity MUST be less than this number. Considering that the traditional best guess at the actual value Graham was looking for was probably 6, this is also one of the worst upper bounds ever devised (although its recently been shown that the number must actually be over 11).
