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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

### What are Knuth up-arrows? (Definition)

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

### How to calculate using Knuth up-arrows notation?

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 1 Knuth up-arrow mean?

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

Example: $4 \uparrow 5 = 4 ^ 5 = 1024$

### What does 3 Knuth up-arrows mean?

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

### Can rational number be used?

No, tetration is only defined for integer numbers.

### 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 written into numbers easily nor 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" source code. Except explicit open source licence (indicated CC / Creative Commons / free), the "Knuth's Arrows" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, 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, copy-paste, or API access for "Knuth's Arrows" are not public, same for offline use on PC, tablet, iPhone or Android ! Remainder : dCode is free to use.

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