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Knuth's up-arrow notation

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Method of notation of very large integers

Inmathematics,Knuth's up-arrow notation is a method of notation forvery large integers, introduced byDonald Knuth in 1976.^[1]

In his 1947 paper,^[2]R. L. Goodstein introduced the specific sequence of operations that are now calledhyperoperations. Goodstein also suggested the Greek namestetration,pentation, etc., for the extended operations beyondexponentiation. The sequence starts with aunary operation (thesuccessor function withn = 0), and continues with thebinary operations ofaddition (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 $H_{n}(a,b)$ .Knuth's up-arrow notation $\uparrow$ is another. For example:

the single arrow $\uparrow$ representsexponentiation (iterated multiplication) $2\uparrow 4=H_{3}(2,4)=2\times (2\times (2\times 2))=2^{4}=16$
the double arrow $\uparrow \uparrow$ representstetration (iterated exponentiation) $2\uparrow \uparrow 4=H_{4}(2,4)=2\uparrow (2\uparrow (2\uparrow 2))=2^{2^{2^{2}}}=2^{16}=65,536$
the triple arrow $\uparrow \uparrow \uparrow$ representspentation (iterated tetration) ${\begin{aligned}2\uparrow \uparrow \uparrow 4&=H_{5}(2,4)\\&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow \uparrow 2))\\&=2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow 2))\\&=2\uparrow \uparrow (2\uparrow \uparrow 4)\\&=\underbrace {2\uparrow (2\uparrow (2\uparrow \cdots ))} \;=\;\underbrace {\;2^{2^{\cdots ^{2}}}} \\&\;\;\;\;\;2\uparrow \uparrow 4{\text{ copies of }}2\;\;\;\;\;{\text{65,536 2s}}\\\end{aligned}}$

The general definition of the up-arrow notation is as follows (for $a\geq 0,n\geq 1,b\geq 0$ ): $a\uparrow ^{n}b=H_{n+2}(a,b)=a[n+2]b.$ Here, $\uparrow ^{n}$ stands forn arrows, so for example $2\uparrow \uparrow \uparrow \uparrow 3=2\uparrow ^{4}3,$ and the square brackets used in the far rhs. expression is another notation for hyperoperations.

Introduction

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Thehyperoperations naturally extend thearithmetic operations ofaddition andmultiplication as follows.Addition by anatural number is defined as iterated incrementation:

{\begin{matrix}H_{1}(a,b)=a+b=&a+\underbrace {1+1+\dots +1} \\&b{\mbox{ copies of }}1\end{matrix}}

Multiplication by anatural number is defined as iteratedaddition:

{\begin{matrix}H_{2}(a,b)=a\times b=&\underbrace {a+a+\dots +a} \\&b{\mbox{ copies of }}a\end{matrix}}

For example,

{\begin{matrix}4\times 3&=&\underbrace {4+4+4} &=&12\\&&3{\mbox{ copies of }}4\end{matrix}}

Exponentiation for a natural power $b {\displaystyle b}$ is defined as iterated multiplication, which Knuth denoted by a single up-arrow:

{\begin{matrix}a\uparrow b=H_{3}(a,b)=a^{b}=&\underbrace {a\times a\times \dots \times a} \\&b{\mbox{ copies of }}a\end{matrix}}

For example,

{\begin{matrix}4\uparrow 3=4^{3}=&\underbrace {4\times 4\times 4} &=&64\\&3{\mbox{ copies of }}4\end{matrix}}

Tetration is defined as iterated exponentiation, which Knuth denoted by a “double arrow”:

{\begin{matrix}a\uparrow \uparrow b=H_{4}(a,b)=&\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} &=&\underbrace {a\uparrow (a\uparrow (\cdots \uparrow a))} \\&b{\mbox{ copies of }}a&&b{\mbox{ copies of }}a\end{matrix}}

For example,

{\begin{matrix}4\uparrow \uparrow 3=&\underbrace {4^{4^{4}}} &=&\underbrace {4\uparrow (4\uparrow 4)} &=&4^{256}&&\\&3{\mbox{ copies of }}4&&3{\mbox{ copies of }}4\end{matrix}}

Expressions are evaluated from right to left, as the operators are defined to beright-associative.

According to this definition,

3\uparrow \uparrow 2=3^{3}=27

3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987

3\uparrow \uparrow 4=3^{3^{3^{3}}}=3^{3^{27}}=3^{7625597484987}

3\uparrow \uparrow 5=3^{3^{3^{3^{3}}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}}

etc.

This already leads to some fairly large numbers, but the hyperoperator sequence does not stop here.

Pentation, defined as iterated tetration, is represented by the “triple arrow”:

{\begin{matrix}a\uparrow \uparrow \uparrow b=H_{5}(a,b)=&\underbrace {a_{}\uparrow \uparrow (a\uparrow \uparrow (\cdots \uparrow \uparrow a))} \\&b{\mbox{ copies of }}a\end{matrix}}

Hexation, defined as iterated pentation, is represented by the “quadruple arrow”:

{\begin{matrix}a\uparrow \uparrow \uparrow \uparrow b=H_{6}(a,b)=&\underbrace {a_{}\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow (\cdots \uparrow \uparrow \uparrow a))} \\&b{\mbox{ copies of }}a\end{matrix}}

and so on. The general rule is that an $n {\displaystyle n}$ -arrow operator expands into a right-associative series of ( $n-1$ )-arrow operators. Symbolically,

{\begin{matrix}a\ \underbrace {\uparrow _{}\uparrow \!\!\cdots \!\!\uparrow } _{n}\ b=\underbrace {a\ \underbrace {\uparrow \!\!\cdots \!\!\uparrow } _{n-1}\ (a\ \underbrace {\uparrow _{}\!\!\cdots \!\!\uparrow } _{n-1}\ (\cdots \ \underbrace {\uparrow _{}\!\!\cdots \!\!\uparrow } _{n-1}\ a))} _{b{\text{ copies of }}a}\end{matrix}}

Examples:

3\uparrow \uparrow \uparrow 2=3\uparrow \uparrow 3=3^{3^{3}}=3^{27}=7,625,597,484,987

{\begin{aligned}3\uparrow \uparrow \uparrow 3&=3\uparrow \uparrow (3\uparrow \uparrow 3)\\&=3\uparrow \uparrow (3\uparrow 3\uparrow 3)\\&={\begin{matrix}\underbrace {3\uparrow 3\uparrow \cdots \uparrow 3} \\3\uparrow 3\uparrow 3{\mbox{ copies of }}3\end{matrix}}\\&={\begin{matrix}\underbrace {3\uparrow 3\uparrow \cdots \uparrow 3} \\{\mbox{7,625,597,484,987 copies of 3}}\end{matrix}}\\&={\begin{matrix}\underbrace {3^{3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}}}} \\{\mbox{7,625,597,484,987 copies of 3}}\end{matrix}}\end{aligned}}

Notation

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In expressions such as $a^{b}$ , the notation for exponentiation is usually to write the exponent $b {\displaystyle b}$ as a superscript to the base number $a {\displaystyle a}$ . But many environments — such asprogramming languages and plain-texte-mail — do not supportsuperscript typesetting. People have adopted the linear notation $a\uparrow b$ for such environments; the up-arrow suggests 'raising to the power of'. If thecharacter set does not contain an up arrow, thecaret (^) is used instead.

The superscript notation $a^{b}$ doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation $a\uparrow b$ instead.

$a\uparrow ^{n}b$ is a shorter alternative notation for n uparrows. Thus $a\uparrow ^{4}b=a\uparrow \uparrow \uparrow \uparrow b$ .

Writing out up-arrow notation in terms of powers

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Attempting to write $a\uparrow \uparrow b$ using the familiar superscript notation gives apower tower.

For example:

a\uparrow \uparrow 4=a\uparrow (a\uparrow (a\uparrow a))=a^{a^{a^{a}}}

If $b {\displaystyle b}$ is a variable (or is too large), the power tower might be written using dots and a note indicating the height of the tower.

a\uparrow \uparrow b={}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{b}

Continuing with this notation, $a\uparrow \uparrow \uparrow b$ could be written with a stack of such power towers, each describing the size of the one above it.

a\uparrow \uparrow \uparrow 4=a\uparrow \uparrow (a\uparrow \uparrow (a\uparrow \uparrow a))=\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{a}}}

Again, if $b {\displaystyle b}$ is a variable or is too large, the stack might be written using dots and a note indicating its height.

a\uparrow \uparrow \uparrow b=\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}b

Furthermore, $a\uparrow \uparrow \uparrow \uparrow b$ might be written using several columns of such stacks of power towers, each column describing the number of power towers in the stack to its left:

a\uparrow \uparrow \uparrow \uparrow 4=a\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow (a\uparrow \uparrow \uparrow a))=\left.\left.\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}a

And more generally:

a\uparrow \uparrow \uparrow \uparrow b=\underbrace {\left.\left.\left.\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {a^{a^{.^{.^{.{a}}}}}} _{\underbrace {\vdots } _{a}}}\right\}\cdots \right\}a} _{b}

This might be carried out indefinitely to represent $a\uparrow ^{n}b$ as iterated exponentiation of iterated exponentiation for any $a {\displaystyle a}$ , $n {\displaystyle n}$ , and $b {\displaystyle b}$ (although it clearly becomes rather cumbersome).

Using tetration

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The Rudy Rucker notation $^{b}a$ fortetration allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call thesetetration towers).

a\uparrow \uparrow b={}^{b}a

a\uparrow \uparrow \uparrow b=\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{b}

a\uparrow \uparrow \uparrow \uparrow b=\left.\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{\underbrace {^{^{^{^{^{a}.}.}.}a}a} _{\underbrace {\vdots } _{a}}}\right\}b

Finally, as an example, the fourth Ackermann number $4\uparrow ^{4}4$ could be represented as:

\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{4}}}=\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{\underbrace {^{^{^{^{^{4}.}.}.}4}4} _{^{^{^{4}4}4}4}}

Generalizations

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Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then ann-arrow operator $\uparrow ^{n}$ is useful (and also for descriptions with a variable number of arrows), or equivalently,hyper operators.

Some numbers are so large that even that notation is not sufficient. TheConway chained arrow notation can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful.

{\begin{matrix}a\uparrow ^{n}b&=&a[n+2]b&=&a\to b\to n\\{\text{(Knuth)}}&&{\text{(hyperoperation)}}&&{\text{(Conway)}}\end{matrix}}

6\uparrow \uparrow 4=\underbrace {6^{6^{.^{.^{.^{6}}}}}} _{4}

, Since

6\uparrow \uparrow 4=6^{6^{6^{6}}}=6^{6^{46,656}}

, Thus the result comes out with

\underbrace {6^{6^{.^{.^{.^{6}}}}}} _{4}

10\uparrow (3\times 10\uparrow (3\times 10\uparrow 15)+3)=\underbrace {100000\ldots 000} _{\underbrace {300000\ldots 003} _{\underbrace {300000\ldots 000} _{15}}}

10^{3\times 10^{3\times 10^{15}}+3}

Even faster-growing functions can be categorized using anordinal analysis called thefast-growing hierarchy. The fast-growing hierarchy uses successive function iteration and diagonalization to systematically create faster-growing functions from some base function $f(x)$ . For the standard fast-growing hierarchy using $f_{0}(x)=x+1$ , $f_{2}(x)$ already exhibits exponential growth, $f_{3}(x)$ is comparable to tetrational growth and is upper-bounded by a function involving the first four hyperoperators;. Then, $f_{\omega }(x)$ is comparable to theAckermann function, $f_{\omega +1}(x)$ is already beyond the reach of indexed arrows but can be used to approximateGraham's number, and $f_{\omega ^{2}}(x)$ is comparable to arbitrarily-long Conway chained arrow notation.

These functions are all computable. Even faster computable functions, such as theGoodstein sequence and theTREE sequence require the usage of large ordinals, may occur in certain combinatorical and proof-theoretic contexts. There exist functions which grow uncomputably fast, such as theBusy Beaver, whose very nature will be completely out of reach from any up-arrow, or even any ordinal-based analysis.

Definition

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Without reference tohyperoperation the up-arrow operators can be formally defined by

a\uparrow ^{n}b={\begin{cases}a^{b},&{\text{if }}n=1;\\1,&{\text{if }}n>1{\text{ and }}b=0;\\a\uparrow ^{n-1}(a\uparrow ^{n}(b-1)),&{\text{otherwise }}\end{cases}}

for all integers $a, b, n {\displaystyle a,b,n}$ with $a\geq 0,n\geq 1,b\geq 0$ .^{[nb 1]}

This definition usesexponentiation $(a\uparrow ^{1}b=a\uparrow b=a^{b})$ as the base case, andtetration $(a\uparrow ^{2}b=a\uparrow \uparrow b)$ as repeated exponentiation. This is equivalent to thehyperoperation sequence except it omits the three more basic operations ofsuccession,addition andmultiplication.

One can alternatively choosemultiplication $(a\uparrow ^{0}b=a\times b)$ as the base case and iterate from there. Thenexponentiation becomes repeated multiplication. The formal definition would be

a\uparrow ^{n}b={\begin{cases}a\times b,&{\text{if }}n=0;\\1,&{\text{if }}n>0{\text{ and }}b=0;\\a\uparrow ^{n-1}(a\uparrow ^{n}(b-1)),&{\text{otherwise }}\end{cases}}

for all integers $a, b, n {\displaystyle a,b,n}$ with $a\geq 0,n\geq 0,b\geq 0$ .

Note, however, that Knuth did not define the "nil-arrow" ( $\uparrow ^{0}$ ). One could extend the notation to negative indices (n ≥ -2) in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:

H_{n}(a,b)=a[n]b=a\uparrow ^{n-2}b{\text{ for }}n\geq 0.

The up-arrow operation is aright-associative operation, that is, $a\uparrow b\uparrow c$ is understood to be $a\uparrow (b\uparrow c)$ , instead of $(a\uparrow b)\uparrow c$ . If ambiguity is not an issue parentheses are sometimes dropped.

Tables of values

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Computing 0↑ⁿ b

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Computing $0\uparrow ^{n}b=H_{n+2}(0,b)=0[n+2]b$ results in

0, whenn = 0 ^{[nb 2]}

1, whenn = 1 andb = 0 ^{[nb 1]}^{[nb 3]}

0, whenn = 1 andb > 0 ^{[nb 1]}^{[nb 3]}

1, whenn > 1 andb is even (including 0)

0, whenn > 1 andb is odd

Computing 2↑ⁿ b

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Computing $2\uparrow ^{n}b$ can be restated in terms of an infinite table. We place the numbers $2^{b}$ in the top row, and fill the left column with values 2. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

Values of $2\uparrow ^{n}b={}$ $H_{n+2}(2,b)={}$ $2[n+2]b={}$ 2 → b → n
b ⁿ	1	2	3	4	5	6	formula
1	2	4	8	16	32	64	$2^{b}$
2	2	4	16	65,536	$2^{65536}\approx 2.00353\times 10^{19728}$	$2^{2^{65536}}\approx 2.12004\times 10^{6.03123\times 10^{19727}}$	$2\uparrow \uparrow b$
3	2	4	65,536	${}^{65536}2=$ 24,636,...,948,736	${}^{{}^{65536}2}2=$ 1,300,...,948,736	${}^{{}^{{}^{65536}2}2}2=$ 320,146,...,948,736	$2\uparrow \uparrow \uparrow b$
4	2	4	${}^{65536}2=$ 24,636,...,948,736	$2\uparrow \uparrow \uparrow ({}^{65536}2)=$ 68,225,...,948,736	167,167,...,948,736	3,449,...,948,736	$2\uparrow \uparrow \uparrow \uparrow b$

The table is the same asthat of the Ackermann function, except for a shift in $n {\displaystyle n}$ and $b {\displaystyle b}$ , and an addition of 3 to all values.

Computing 3 ↑ⁿ b

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We place the numbers $3^{b}$ in the top row, and fill the left column with values 3. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

Values of $3\uparrow ^{n}b={}$ $H_{n+2}(3,b)={}$ $3[n+2]b={}$ 3 → b → n
b ⁿ	1	2	3	4	5	formula
1	3	9	27	81	243	$3^{b}$
2	3	27	7,625,597,484,987	$3^{7625597484987}\approx 1.25801\times 10^{3638334640024}$	$3^{3^{7625597484987}}=$ 338,605,...,355,387	$3\uparrow \uparrow b$
3	3	7,625,597,484,987	1,945,...,195,387	93,652,...,195,387	4,854,...,195,387	$3\uparrow \uparrow \uparrow b$
4	3	1,945,...,195,387	834,215,...,195,387	25,653,...,195,387	17,124,...,195,387	$3\uparrow \uparrow \uparrow \uparrow b$

Computing 4 ↑ⁿ b

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We place the numbers $4^{b}$ in the top row, and fill the left column with values 4. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

Values of $4\uparrow ^{n}b={}$ $H_{n+2}(4,b)={}$ $4[n+2]b={}$ 4 → b → n
b ⁿ	1	2	3	4	5	formula
1	4	16	64	256	1024	$4^{b}$
2	4	256	$2^{512}\approx 1.34078\times 10^{154}$	$2^{2^{513}}\approx 2.36102\times 10^{8.07230\times 10^{153}}$	$2^{2^{2^{513}+1}}\approx 10^{10^{8.07230\times 10^{153}}}$	$4\uparrow \uparrow b$
3	4	$2^{2^{513}}\approx 2.36102\times 10^{8.07230\times 10^{153}}$	${}^{2^{2^{513}}}4$	${}^{{}^{2^{2^{513}}}4}4$	$4\uparrow \uparrow \uparrow 5$	$4\uparrow \uparrow \uparrow b$
4	4	${}^{{}^{2^{2^{513}}}4}4$	807,230,472,602,822,537,... << 118 >> ...,481,244,990,261,351,117	10^10¹⁵³ digits	10^{10^10¹⁵³} digits	$4\uparrow \uparrow \uparrow \uparrow b$

Computing 10 ↑ⁿ b

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We place the numbers $10^{b}$ in the top row, and fill the left column with values 10. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

Values of $10\uparrow ^{n}b={}$ $H_{n+2}(10,b)={}$ $10[n+2]b={}$ 10 → b → n
b ⁿ	1	2	3	4	5	formula
1	10	100	1,000	10,000	100,000	$10^{b}$
2	10	10,000,000,000	$10^{10,000,000,000}$	$10^{10^{10,000,000,000}}$	$10^{10^{10^{10,000,000,000}}}$	$10\uparrow \uparrow b$
3	10	${\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}$	${\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}$	${\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}$	${\begin{matrix}\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\\underbrace {10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}} \\10{\mbox{ copies of }}10\end{matrix}}$	$10\uparrow \uparrow \uparrow b$
4	10	${\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}$	${\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}$	${\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}$	${\begin{matrix}\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\\underbrace {^{^{^{^{^{10}.}.}.}10}10} \\10{\mbox{ copies of }}10\end{matrix}}$	$10\uparrow \uparrow \uparrow \uparrow b$

For 2 ≤b ≤ 9 the numerical order of the numbers $10\uparrow ^{n}b$ is thelexicographical order withn as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line. The same applies for the numbers in the 97 columns with 3 ≤b ≤ 99, and if we start fromn = 1 even for 3 ≤b ≤ 9,999,999,999.

Notes

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^^a ^b ^cFor more details, seePowers of zero.
^Keep in mind that Knuth did not define the operator $\uparrow ^{0}$ .
^^a ^bFor more details, seeZero to the power of zero.

References

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^Knuth, Donald E. (1976). "Mathematics and Computer Science: Coping with Finiteness".Science.194 (4271):1235–1242.Bibcode:1976Sci...194.1235K.doi:10.1126/science.194.4271.1235.PMID 17797067.S2CID 1690489.
^R. L. Goodstein (Dec 1947). "Transfinite Ordinals in Recursive Number Theory".Journal of Symbolic Logic.12 (4):123–129.doi:10.2307/2266486.JSTOR 2266486.S2CID 1318943.

External links

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Weisstein, Eric W."Knuth Up-Arrow Notation".MathWorld.
Robert Munafo,Large Numbers: Higher hyper operators

v t e Hyperoperations
Hyper-operators:	Successor Addition Multiplication Exponentiation Tetration Pentation Hexation Heptation Octation Ennation Dekation (en/do/tria) Icosation (do) Triantation Hectation (en) Dohectation Chiliation Myriation Faxulation (en/dy) Faxulyriation Circulation
Bowers's extensions:	Expansion (muiti/power/expando) Explosion (multi/power/expando) Detonation (multi/power) Pentonation (multi)
Username's extensions:	Hexonation Heptonation Octonation Ennonation Dekonation
Tiaokhiao's extensions:	Megotion (muiti/power/tetra) Megoexpansion (multi/power) Megoexplosion (multi) Megodetonation Gigotion (expand/explod/deto) Terotion (expand) Petotion Exotion Zettotion Yottotion
Saibian's extensions:	Powiaination (expand/explod/deto) Megodaination (expand/explod) Gigodaination (expand) Terodaination Powiairation (megod/gigod/terod) Powiaintation (megod) Powiairtation

Large numbers

Examples
in
numerical
order

Expression
methods

Notations	Scientific notation Knuth's up-arrow notation Conway chained arrow notation Steinhaus–Moser notation
Operators	Hyperoperation Tetration Pentation Ackermann function Grzegorczyk hierarchy Fast-growing hierarchy Slow-growing hierarchy Hardy hierarchy Veblen function

v t e Donald Knuth
Publications	The Art of Computer Programming "The Complexity of Songs" Computers and Typesetting Concrete Mathematics Surreal Numbers Things a Computer Scientist Rarely Talks About Selected papers series
Software	TeX Metafont MIXAL (MIX MMIX)
Fonts	AMS Euler Computer Modern Concrete Roman
Literate programming	WEB CWEB
Algorithms	Knuth's Algorithm X Knuth–Bendix completion algorithm Knuth–Morris–Pratt algorithm Knuth shuffle Robinson–Schensted–Knuth correspondence Trabb Pardo–Knuth algorithm Generalization ofDijkstra's algorithm Knuth's Simpath algorithm
Other	Dancing Links Knuth reward check Knuth Prize Knuth's up-arrow notation Man or boy test Quater-imaginary base -yllion Potrzebie system of weights and measures

Movatterモバイル変換

Introduction

Notation

Writing out up-arrow notation in terms of powers

Using tetration

Generalizations

Definition

Tables of values

Computing 0↑n b

Computing 2↑n b

Computing 3 ↑n b

Computing 4 ↑n b

Computing 10 ↑n b

See also

Notes

References

External links

Computing 0↑ⁿ b

Computing 2↑ⁿ b

Computing 3 ↑ⁿ b

Computing 4 ↑ⁿ b

Computing 10 ↑ⁿ b