The notation is defined by

- (
*m*number of*n*s) - with
*m*number of*n*s on the RHS. - with
*m*number of*n*s on the RHS.

It is similar to standard exponentiation in a sense; for example:

The up-arrow notation gets cumbersome when many arrows are needed (eg Graham's number). The hyper operators and Conway chained arrow offer clearer alternatives: hyper*(n,p+2,m) = n→m→p = n^…^m* with *p* arrows. It is generally suggested that Knuth's arrow should be used for relatively smaller magnitude numbers, and the chained arrow or hyper operators for larger ones.

The numbers involved are staggeringly large. Take as an example the very small and simple case:

When foolhardily trying to evaluate on a computer, 8^8 evaluates as 16777216 but 8^16777216 generates a warning message that the calculation will take a very long time. That step never finished, let alone the fourth step! In fact, this number has more digits than there are atoms in the visible universe, by 15,151,256 orders of magnitude.