First of all, to understand large numbers, we need to understand this (up arrow) notation. This notation was introduced by Donald Knuth in 1976. The idea behind this notation is that this notation is based on the fact that multiplication as the iterated addition, and exponentiation as the iterated multiplication. Continuing it produces tetration, and so on. It is based on the hyper-operator. The symbol that is used is "↑" or "^".
a↑b = a^b (or a to the power of b)
a↑(n)1 = a
a↑(n)0 = 1
Note: a↑(n) b is an abbreviation of a↑↑↑↑↑....↑↑↑↑↑b (where there are n arrows).
This operator is right-associative, so we need to solve the operation from right to left, unlike down-arrow, which is left-associative, which solves the other way around.
For example, 3↑3↑3 = 3↑(3↑3) = 3^27 = 7,625,597,484,987, but (3^3)^3 solves into 27^3, which is 19,683. (discussion later)
Hyper-operator equivalent to (up) arrow notation:
exponentiation = a^b
tetration = a↑↑b = a^a^a^a...a^a^a (where there are b a's) (repeated exponentiation)
pentation = a↑↑↑b = a^^a^^a^^a...a^^a^^a^^a (where there are b a's) (repeated tetration)
hexation = a↑↑↑↑b = a^^^a^^^a^^^a...a^^^a^^^a (where there are b a's) (repeated pentation)
etc.
Generally, (n+2)ation = a↑(n)b = a↑(n-1)a↑(n-1)a...a↑(n-1)a↑(n-1)a (where there are b a's).
Arrow notation can also be defined in other notations. For example:
In BEAF and Bird's array notation, both can be expressed as a{n}b (in case of the former) or {a,b,n} (n means no. of arrows to re-iterate)
In chained arrow notation (which will be discussed separately), it is expressed as a→b→n.
In strong array notation, it is similar to no. 1, except it uses s() rather than {}.
(WIP)
Here are some examples of the usage of up arrow notation:
3↑3 = 3*3*3 = 3*9 = 27
2↑32 = 2*2*2*2...2*2*2*2 (where there are 32 2's) = 4,294,967,296
2↑↑↑2 = 2↑↑2 =2↑2 = 4 (as a result, 2{n}2, regardless of n's, still equals to 4)
10↑100 = 10^100 (which is googol)
2↑↑↑3 = 2↑↑2↑↑2 = 2↑↑4 = 2↑2↑2↑2 = 65,536
3↑↑↑3 = 3↑↑3↑↑3 = 3↑↑(3↑3↑3) = 3↑↑(3↑27) = 3↑↑7,625,597,484,987 = 3↑3↑3↑3...3↑3↑3 (where there are 7,625,597,484,987 3's) (already LARGE)
As the name would suggests, this notation is obviously weaker than their up arrow counterparts. It is a left associative extension of addition, multiplication, etc. (ie. solving from left to right).
a↓b = a^b (similar to a↑b)
a↓(n)1 = a
a↓(n)(b+1) = (a↓(n)b)↓(n-1)a
It looks like:
n↓↓n = n^n^(n-1) < n↑↑3
This inequality is helpful when bounding down arrows in terms of up arrows. Also, it is proven that:
a↓(2n-1)b ≥ a↑(n)b, (for a, n, b ≥ 1)
To understand this notation even more, I will provide some examples, even though it is not as important as the up-arrow counterpart.
10↓10 = 10^10
10↓↓3 = 10↓10↓10 = (10↓10)↓10 = (10^10)^10 = 10^(10*10) = 10^100 (solve from left to right)
10↓↓4 = 10↓10↓10↓10 = (10↓↓3)↓10 = (10^100)^10 = 10^(100*10) = 10^1000 = 10^10^3
10↓↓10 = 10↓↓↓2 = 10^10^9
10↓↓11 = 10↑↑3
10↓↓↓3 = (10↓↓10)↓↓10 ≈ 10↑↑4
So, with the understanding of these notations, we can now proceed to the next topic.