Knuth's Arrows

Knuth arrows (or Knuth arrow operators) are a set of mathematical symbols dedicated to the representation of repeated exponentiation (iterated powers or tetration).

$$ a \uparrow ^ n b = \begin{cases} a^b & n=1 \\ 1 & n > 1 \ \& \ b = 0 \\ a \uparrow ^ {n-1} (a \uparrow ^{n} (b-1)) & \end{cases} $$

As multiplication is the repetition of additions ($ 2 \times 3 = 2+2+2 $), as exponentiation is the repetition of multiplications ($ 2^3 = 2 \times 2 \times 2 $), the knuth arrows is the repetition of exponentiations (also called iterated exponentiation or tetration).

Knuth's notation with a single arrow represents a simple power operation (a single arrow represents an exponentiation)

Knuth's notation with 2 arrows is an iterated power

$$ a \uparrow \uparrow b = \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}}_{b} $$

It may be noted that,

$$ a \uparrow\uparrow b = \underbrace{a_{}\uparrow a\uparrow\dots\uparrow a}_{b} $$

The notation with 1 arrow represents a simple exponentiation (a power, an exponent).

The 3 arrows (triple arrow) notation is the continuity of the 2 arrows notation (double arrow)

$$ a \uparrow\uparrow\uparrow b = \underbrace{a_{}\uparrow\uparrow a\uparrow\uparrow\dots\uparrow\uparrow a}_{b} $$

No, tetration is only defined for integer numbers.

No need to try with decimal numbers, the decimal point will be ignored.

Knuyth's arrows make it possible to represent numbers so large that the usual notations do not allow them to be written into numbers easily nor precisely.

Knuth arrows are generally implemented by recursion in code:// pseudo-code
function knuthArrows(a, n, b) {
if (b == 0) return 1
if (n == 1) return a ** b
return knuthArrows(a, n-1, knuthArrows(a, n, b - 1))
}