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Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 50,930

Knuth's up-arrow notation

In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.

In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc.

Various notations have been used to represent hyperoperations. One such notation is

. Another notation is

, an infix notation which is convenient for ASCII. The notation

is known as 'square bracket notation'.

Knuth's up-arrow notation

  is an alternative notation. It is obtained by replacing

in the square bracket notation by

arrows.

For example:

the single arrow

  represents exponentiation (iterated multiplication)

the double arrow

  represents tetration (iterated exponentiation)

the triple arrow

represents pentation (iterated tetration)

The general definition of the up-arrow notation is as follows (for

:

Here,

stands for n arrows, so for example

.

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hypsin_0

Member

From: China,Shenzhen

Registered: 2025-07-05

Posts: 20

Re: Knuth's up-arrow notation

Also try:Linear array notation(a  kind of notation)
1 {a,1,...}=a
2 {a,b,1,...}=a^b
3 {...,1}={...}
The first item is called"base",the second is called"power",after power,the first item(not 1) called"pilot",the item before pilot is
called"copilot".
4 If we can't use rule 1,2,3 ,change pilot into pilot-1,change copilot into the array,but change the power into power-1.

For example
{3,2,3}={3,{3,1,3},2}={3,3,2}={3,{3,2,2},1}=......=3^27

{a,b,c}=a[b+2]c

let f(n)=n[n]n
     g(n)={n,n,n,......,n}(n's n)
f(n)'s fgh(a googology term) is only ω
but g(n) has ω^ω

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