Extensions to Arrow Notation

In the last article we covered hyperoperations, which are expressed using up-arrow notation. In this article we will cover the various extensions to up-arrow notation.

Excelguru's Extended Arrow Notation

On a forum elsewhere on the internet, a user made an extension to up-arrow notation, which is defined as follows:

a{^}b = a^^^...^^^a w/b ^s

a{^)^b = a{^}a{^}a...a{^}a{^}a w/b a's

1{^}(any number) is equal to 1 and 2{^}(any number) is equal to 4. So we will begin with:

3{^}1 = 3^3 = 3³ = 27

3{^}2 = 3^^3 = 3^3^3 = 3²⁷ = 7,625,597,484,987

3{^}3 = 3^^^3 = 3^^(3^^3) = 3^^7,625,597,484,987

3{^}4 = 3^^^^3 = 3^^^(3^^^3) = 3^^^(3^^7,625,597,484,987) = 3^^(3^^(3^^(...(3^^(3^^3))...))) w/3^^7,625,597,484,987 3s

3{^}5 = 3^^^^^3 = 3^^^^(3^^^^3) = 3^^^^(3^^^(3^^^3)) = 3^^^^(3^^^(3^^7,625,597,484,987))

3{^}6 = 3^^^^^^3 = 3^^^^^(3^^^^^3) = 3^^^^^(3^^^^(3^^^^3))

4{^}1 = 4^4 = 4⁴ = 256

4{^}2 = 4^^4 = 4^4^4^4 = 4^4^256 ~ 10^{8.0723*10^153}

4{^}3 = 4^^^4 = 4^^(4^^(4^^4))

4{^}4 = 4^^^^4 = 4^^^(4^^^(4^^^4))

3{^}^1 = 3

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

3{^}^3 = 3{^}3{^}3 = 3{^}(3^^7,625,597,484,987) = 3^^^^^^^^^^...^^^^^^^^3 (w/ 3^^^3 ^'s)

To prevent the degeneration to 4 for base 2, I would suggest simply changing the definition like so:

a{^}b = a^^^^^...^^^^^a w/ b ^s, a >= 3

2{^}b = 2^^^^^...^^^^^3 w/ b ^s

We can then have:

2{^}1 = 2³ = 8

2{^}2 = 2^^3 = 2⁴ = 16

2{^}3 = 2^^^3 = 2^^4 = 65,536

2{^}4 = 2^^^^3 = 2^^^4 = 2^^65,536

2{^}^2 = 2{^}2 = 16

2{^}^3 = 2{^}16 = 2^^^^^^^^^^^^^^^^3

In 1995, J. H. Conway and R. K. Guy defined Conway chained arrow notation (which we will cover in more detail in a later article), which is defined like so:

a -> b = a^b

a -> b -> c = a^^...^^b with c ^s

a -> b -> c -> d = a -> b -> (a -> b -> c-1 -> d) -> d-1

Similar to power towers, if any number in a chained expression is 1, it and all further numbers are ignored.

Here are some examples of chained expressions with 3 terms:

2 -> 3 -> 4 = 2^^^^3 = 2^^^4 = 2^^65,536

3 -> 2 -> 3 = 3^^^2 = 3^^3 = 3^3^3 = 3²⁷ = 7,625,597,484,987

5 -> 4 -> 3 = 5^^^4 = 5^^(5^^(5^^5))