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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

## Answers to Questions (FAQ)

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

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).

### 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 (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.

### 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.

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

### 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.

### How to implement knuth arrows?

Knuth arrows are generally implemented by recursion in code:// pseudo-codefunction 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))}

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Knuth's Arrows on dCode.fr [online website], retrieved on 2024-09-10, https://www.dcode.fr/knuth-arrows

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