# Knuth's up-arrow notation

The 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:

*m* ↑ *n* = *m* × *m* × ... × *m* (*n* terms) = *m*^{n}

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 = 3^{7625597484987}

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.