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This article describes
Knuth's up-arrow notation to represent
iterated exponentiation with base
(
power towers with base
) and a
down-arrow notation to represent the
iterated logarithm (with base
) inverse of
iterated exponentiation.
Knuth's up-arrow notation
In 1976, Donald Knuth introduced the following up-arrow notation for power towers
![{\displaystyle p=b\uparrow ^{n}d=b\uparrow ^{n-1}[b\uparrow ^{n}(d-1)],\quad n\geq 2,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/4d36f9f152448b4b433c9912746830690aab1b3c)
with
![{\displaystyle b\uparrow ^{1}d:=b^{d},\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d7697fd6a18ac3bb0cb94db7c053c33bfa796fde)
![{\displaystyle b\uparrow ^{n}0:=1,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9e891d3d8052e7e38bc9680d1707a20038fb92ca)
where
is the order of the power tower with base
, and
is the result of the power tower evaluation.
Down-arrow notation
The down-arrow notation defines a logarithmic-type inverse of the up-arrow notation
![{\displaystyle d=b\downarrow ^{n}p:=\log _{b}^{*}p,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3e963fc1b53eac606c93a456d5fab223ea935036)
where
is the number of
iterations of the logarithm (base
) required such that
.
In particular, we have
![{\displaystyle d=e\downarrow ^{n}p:=\log _{e}^{*}p=\log ^{*}p,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/8d26922302cb56c91b37aa89b2eb65abfc520d16)
where
is the number of
iterations of the natural logarithm required such that
.
Now, if we have
![{\displaystyle p=b\uparrow ^{n}d,\quad d\geq 1,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/2bf9ef27c1561a58443f66d36a2932dbe5aa6a1f)
then
![{\displaystyle d=b\downarrow ^{n}p,\,}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/372e72b5c516bdd05f71383cc65b28d738bf0df4)
where
is a
positive integer. How could we generalize
to
integers,
rational numbers,
real numbers or
complex numbers for all
? For
, we do have a generalization.
See also
Hierarchical list of operations pertaining to numbers [1] [2]
0th iteration
1st iteration
- Addition:
S(S(⋯ "a times" ⋯ (S(n)))) |
, the sum , where is the augend and is the addend. (When addition is commutative both are simply called terms.)
- Subtraction:
P(P(⋯ "s times" ⋯ (P(n)))) |
, the difference , where is the minuend and is the subtrahend.
2nd iteration
- Multiplication:
n + (n + (⋯ "k times" ⋯ (n + (n)))) |
, the product , where is the multiplicand and is the multiplier.[3] (When multiplication is commutative both are simply called factors.)
- Division: the ratio , where is the dividend and is the divisor.
3rd iteration
- Exponentiation ( as "degree", as "base", as "variable").
- Powers:
n ⋅ (n ⋅ (⋯ "d times" ⋯ (n ⋅ (n)))) |
, written .
- Exponentials:
b ⋅ (b ⋅ (⋯ "n times" ⋯ (b ⋅ (b)))) |
, written .
- Exponentiation inverses ( as "degree", as "base", as "variable").
4th iteration
- Tetration ( as "degree", as "base", as "variable").
- Tetration inverses ( as "degree", as "base", as "variable").
5th iteration
- Pentation ( as "degree", as "base", as "variable").
- Pentation inverses
6th iteration
- Hexation ( as "degree", as "base", as "variable").
- Hexation inverses
7th iteration
- Heptation ( as "degree", as "base", as "variable").
- Heptation inverses
8th iteration
- Octation ( as "degree", as "base", as "variable").
- Octa-powers:
n ^^^^^ (n ^^^^^ (⋯ "d times" ⋯ (n ^^^^^ (n)))) |
, written .
- Octa-exponentials:
b ^^^^^ (b ^^^^^ (⋯ "n times" ⋯ (b ^^^^^ (b)))) |
, written .
- Octation inverses
Notes
External links