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Knuth's Arrows

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.

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Knuth's Arrows -

Tag(s) : Arithmetics, Notation System

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Knuth's Arrows

Calculation with Knuth's up-arrows notation A↑↑B



Answers to Questions (FAQ)

How to calculate using Knuth up-arrows notation?

The Knuth arrows are a repeated exponentiation representation (Knuth up arrows). 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).

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 (big integers) that dCode can not display without risking blocking your browser, so there's an automatic limit above several thousands of digits.

What does 3 Knuth up-arrows means?

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

Why using Knuth up-arrows?

Knuyth's arrows make it possible to represent numbers so large that the usual notations do not allow them to be described simply or precisely.

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

Source code

dCode retains ownership of the online 'Knuth's Arrows' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Knuth's Arrows' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Knuth's Arrows' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Knuth's Arrows' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.

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Questions / Comments

Thanks to your feedback and relevant comments, dCode has developed the best 'Knuth's Arrows' tool, so feel free to write! Thank you!


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