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

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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).

**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.

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.

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.

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

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

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