A

David

Darling

Knuth's up-arrow notation

Knuth's up-arrow notation is a notation for large numbers developed by the American mathematician Donald Knuth (1938–) in 1976. A single up-arrow (↑) is the same as exponentiation:

 

mn = m × m × ... × m (n terms) = mn

 

Two up-arrows together represent a power tower: m ↑↑ n = m m^m^...^m (a tower of height n), which is the same as the operation known as hyper4 or tetration. This can very rapidly generate huge numbers. For example:

 

2 ↑↑ 2 = 2 ↑ 2 = 4
2 ↑↑ 3 = 2 ↑ 2 ↑ 2 = 2 ↑ 4 = 16
2 ↑↑ 4 = 2 ↑ 2 ↑ 2 ↑ 2 = 2 ↑ 16 = 65536
3 ↑↑ 2 = 3 ↑ 3 = 27
3 ↑↑ 3 = 3 ↑ 3 ↑ 3 = 3 ↑ 27 = 7625597484987
3 ↑↑ 4 = 3 ↑ 3 ↑ 3 ↑ 3 = 3 ↑ 3 ↑ 27 = 37625597484987

 

Three up-arrows together represent a still more vastly powerful operator, equivalent to hyper5 or pentation, or a power tower of power towers:

 

m ↑↑↑ n = m ↑↑ m ↑↑...↑↑ m (n terms).

 

For example:

 

2 ↑↑↑ 2 = 2 ↑↑ 2 = 4
2 ↑↑↑ 3 = 2 ↑↑ 2 ↑↑ 2 = 2 ↑↑ 4 = 65 536
2 ↑↑↑ 4 = 2 ↑↑ 2 ↑↑ 2 ↑↑ 2 = 2 ↑↑ 65536 = 2 ↑ 2 ↑...↑ 2 (65 536 terms)
3 ↑↑↑ 2 = 3 ↑↑ 3 = 7 625 597 484 987
3 ↑↑↑ 3 = 3 ↑↑ 3 ↑↑ 3 = 3 ↑↑ 7 625 597 484 987 = 3 ↑ 3 ↑...↑ 3 (a power tower 7625597484987 layers high)
3 ↑↑↑ 4 = 3 ↑↑ 3 ↑↑ 3 ↑↑ 3 = 3 ↑↑ 3 ↑↑ 7625597484987 = 3 ↑↑ 3 ↑...↑ 3 (a tower 3 ↑↑ 7625597484987 layers high)

 

Similarly,

 

m ↑↑↑↑ n = m ↑↑↑ m ↑↑↑...↑↑↑ m (n terms)

 

so that, for example:

 

2 ↑↑↑↑ 2 = 2 ↑↑↑ 2 = 4
2 ↑↑↑↑ 3 = 2 ↑↑↑ 2 ↑↑↑ 2 = 2 ↑↑↑ 4 = 2 ↑ 2 ↑...↑ 2 (65536 terms)
2 ↑↑↑↑ 4= 2 ↑↑↑ 2 ↑↑↑ 2 ↑↑↑ 2 = 2 ↑↑↑ 2 ↑ 2 ↑...↑ 2 (65536 terms)
3 ↑↑↑↑ 2 = 3 ↑↑↑ 3 = 3 ↑ 3 ↑...↑ 3 (7625597484987 terms)
3 ↑↑↑↑ 3 = 3 ↑↑↑ 3 ↑↑↑ 3 = 3 ↑↑↑ 3 ↑ 3 ↑...↑ 3 (7625597484987 terms)
= 3 ↑↑ 3 ↑ 3 ↑...↑ 3 (3 ↑ 3 ↑...↑ 3 (7625597484987 terms) terms)

 

Even up-arrow notation becomes cumbersome, however, when faced with staggeringly large numbers such as Graham's number. For such cases, more extensible systems such as Conway's chained arrow notation or Steinhaus-Moser notation are better suited. See also the Ackermann function, to which up-arrow notation is closely related.