Search for a tool
Knuth's Arrows

Tool to write with Arrowed notation of iterative exponentiation by Knuth : a mathematical notation whose purpose is to write huge integer numbers with powers.

Results

Knuth's Arrows -

Tag(s) : Mathematics, Notation System

dCode and you

dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day!
You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? Contact-me!

Team dCode read all messages and answer them if you leave an email (not published). It is thanks to you that dCode has the best Knuth's Arrows tool. Thank you.

# Knuth's Arrows

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

Tool to write with Arrowed notation of iterative exponentiation by Knuth : a mathematical notation whose purpose is to write huge integer numbers with powers.

## Answers to Questions

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

The Knuth arrows are a repeated exponentiation representation. As multiplication is the iteration of additions ( $$2 \times 3 = 2+2+2$$ ), as exponentiation is the iteration of multiplications (\ = 2 \ times 2 \ times 2 \)), the iterated exponentiation is the repeatition of exponentiations.

Knuth's notation with a single arrow represents a simple power operation

Example: $$3 \uparrow 3 = 3^3 = 27$$

Knuth's notation with 2 arrows is an iterated power

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

Example: $$3 \uparrow\uparrow 2 = 3^3 = 27$$

Example: $$3 \uparrow\uparrow 3 = 3^{3^3} = 3^{27} = 7625597484987$$

Example: $$3 \uparrow\uparrow 4 = 3^{3^{3^3}} = 3^{3^{27}} = 3^{7625597484987}$$

It may be noted that

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

Example: $$3 \uparrow\uparrow 2 = 3 \uparrow 3$$

Example: $$3 \uparrow\uparrow 3 = 3 \uparrow 3 \uparrow 3$$

It is the same with the notation with 3 arrows

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

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

Knuth's arrows produce immensely large numbers that dCode can not afford to display without risking blocking your browser, so there's a limit of 100,000 digits.

## Source code

dCode retains ownership of the source code of the script Knuth's Arrows. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, etc.) which dCode owns rights can be transferred after sales quote. So if you need to download the Knuth's Arrows script for offline use, for you, your company or association, see you on contact page !

## Questions / Comments

Team dCode read all messages and answer them if you leave an email (not published). It is thanks to you that dCode has the best Knuth's Arrows tool. Thank you.

Source : http://www.dcode.fr/knuth-arrows
© 2017 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode