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= Introduction =
The up-arrow notation defined as x↑ʸz, definitions are:
# x↑ʸ2 = x↑ʸ⁻¹x
# x↑ʸ1 = x
# 1↑ˣy = 1
# x↑ʸ0 = 1
# x↑ʸz = x↑ʸ⁻¹[x↑ʸ(z - 1)]
So, hyper(x,y,z) = x↑ᶻ⁻²y. The ackermann function is called A(x,y) = 2↑ʸ⁻²(x + 3) - 3.
The hyperoperators are called:
Succession, Addition, Multiplication, Exponentiation, Tetration, Pentation etc.
See also Chained-arrow notation.
So, we start with addiction, multiplication, exponentiation, the graphing called:
x + y = x + 1 + 1 + ……… + 1 with y 1’s
x × y = x + x + x + ……… + x with y x’s
xʸ = x × x × x × ……… × x with y x’s
etc.
The tetration defined as:
ʸx = x^x^x^………^x with y x’s
x → y → z = x↑ᶻy = the hyperoperation, or hyper(x,z + 2,y).
And the pentation defined as:
x↑³y = x↑↑(x↑↑(x↑↑(………(x↑↑x)………))) with y x’s.
Hexation defined as:
x↑⁴y = x↑³(x↑³(x↑³(………(x↑³x)………))) with y x’s
Heptation defined as:
x↑⁵y = x↑⁴(x↑⁴(x↑⁴(………(x↑⁴x)………))) with y x’s
And hyperoperation defined as:
x↑ᶻy = x↑ᶻ⁻¹(x↑ᶻ⁻¹(x↑ᶻ⁻¹(………(x↑ᶻ⁻¹x)………))) with y x’s.
To continue this, go to Extended up arrow notation.
== Issues ==
The common issue occurs at x↑∞y = infinitation, that is mostly I’ll defined. Since infinity minus 1 is still infinity, infinite loop occurs at infinitation and there are only 4 well defined:
x infinitated to the 1.
1 infinitated to the x.
0 infinitated to the x.
x infinitated to the 0.
== Infinity ==
Infinity is not a number. Even if you add, multiply, exponent, there are computable functions, and even if you put Uncomputable function to infinity, it is still infinity and there is only 1 exception - uncountable infinities are uncountably bigger. ∞0 is 1, ∞1 is ∞, ∞x+1 is x-th infinity fixed point, ∞2 is 1st infinity fixed point - uncountable infinity.
To continue this, go to infinity.
= Operators =
== Tetration ==
This is the first hyperoperator what not appearing in mathematics. Graph below:
Inability to make tetration in graphing calculators make the tetration difficult to graph. You can try these of below 2 workarounds to create tetration. Some of workarounds may not work until these get implemented.
=== Workaround ===
This workaround is not possible without an algorithms, first workaround is called:
logx(y + 2) if -2 < y < -1
y + 1 if -1 < y < 0
xy if 0 < y < 1
xx^(y-1) if 1 < y < 2
xx^x^(y-2) if 2 < y < 3
etc.
There is an example if y and x is 2:
22^2^(2-2) = 22^2^0 = 22^1 = 22 = 4
Second workaround is:
List: Tetration = [-∞,0,1,x,xx,xx^x,…]
The workaround can be defined as Tetration[y + 3] = yx.
There is an example:
The example of y = 2 and x = 3:
Tetration[2 + 3] = Tetration[5],
If x = 3, Tetration[5] = 33 = 27.
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