Knuth's up-arrow notation

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) = 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 = 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.

Related category

• SYMBOLS AND NOTATION

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