Tool to write with Arrowed notation of iterative exponentiation by Knuth: a mathematical notation with arrows aiming to write huge integer numbers with repeated powers.

Knuth's Arrows - dCode

Tag(s) : Arithmetics, Notation System

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

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)

__Example:__ $$ 3 \uparrow 3 = 3^3 = 27 $$

Knuth's notation with 2 arrows is an iterated power

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

__Example:__ $$ 3 \uparrow\uparrow 2 = 3^3 = 27 \\ 3 \uparrow\uparrow 3 = 3^{3^3} = 3^{27} = 7625597484987 \\ 3 \uparrow\uparrow 4 = 3^{3^{3^3}} = 3^{3^{27}} = 3^{7625597484987} $$

It may be noted that,

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

__Example:__ $$ 3 \uparrow\uparrow 2 = 3 \uparrow 3 = 3^3 \\ 3 \uparrow\uparrow 3 = 3 \uparrow 3 \uparrow 3 = 3^{3^3} $$

Knuth's arrows produce *immensely large numbers* (very large integers), so large that they cannot be represented (larger than the memory space available to your browser, or even larger than the number of atoms in the universe). dCode will not perform calculations beyond a few thousand digits.

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

__Example:__ $ 4 \uparrow 5 = 4 ^ 5 = 1024 $

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

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

__Example:__ $$ 3 \uparrow\uparrow\uparrow 3 = 3 \uparrow\uparrow(3 \uparrow\uparrow 3) = 3 \uparrow\uparrow( 3 \uparrow 3 \uparrow 3) $$

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.

The dCode calculator is therefore limited, because the numbers of iterations quickly exceed the capacities of the computers.

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))

}

dCode retains ownership of the "Knuth's Arrows" source code. Except explicit open source licence (indicated Creative Commons / free), the "Knuth's Arrows" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Knuth's Arrows" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Knuth's Arrows" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

Reminder : dCode is free to use.

The copy-paste of the page "Knuth's Arrows" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!

Exporting results as a .csv or .txt file is free by clicking on the *export* icon

Cite as source (bibliography):

*Knuth's Arrows* on dCode.fr [online website], retrieved on 2024-09-10,

knuth,arrow,tetration,exponentiation,power,exponent,donald,notation,big,number,integer,iteration,up,recursion,recursivity

https://www.dcode.fr/knuth-arrows

© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.

Feedback