# Conway's chained-arrow notation

Conway's chained-arrow notation is one of various methods that have been devised recently for representing extremely large numbers. Developed by John Conway, it is based on Knuth's up-arrow notation but is even more powerful. The two systems are related thus:

*a* → *b* → 1 = *a* ↑ *b*

*a* → *b* → 2 = *a* ↑↑ *b*

*a* → *b* → 3 = *a* ↑↑↑ *b*↑↑↑...↑↑ *b* (*c* up arrows)

Longer chains are evaluated by the following general rules:

*a* → ... → *b* → *c* → 1 = *a* → ... →* b* → *c*

*a* → ... → *b* → 1 → *d* + 1 = *a* → ... → *b*, and

*a* → ... → *b* → *c* + 1 → *d* + 1 = *a* → ... → *b* → (*a* → ... → *b* → *c* → *d* + 1) → *d*

It's important to recognize that the Conway arrow isn't an ordinary **dyadic** operator. Where three or more numbers are joined by arrows, the arrows don't
act separately but rather the whole chain has to be considered as a unit.
The chain might be thought of as a function with a variable number of arguments, or as a function whose single argument
is an ordered list or vector. The Ackermann
function is equivalent to a three-element chain: *A*(*m*, *n*) = ( 2 → (*n* + 3) → (*m* - 2)) - 3. It can also be shown that Graham's
number is bigger than 3 → 3 → 64 → 2 and smaller than 3 → 3 → 65 → 2.