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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.

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Knuth's arrows -

Tag(s) : Mathematics,Notation System

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# Knuth's arrows

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

Also on dCode: Exponentiation

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

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

As multiplication is iterative addition, as exponentiation is iterative multiplication, knuth notation is iterative exponentiation.

With only one up-arrow, it is an exponentiation

$$3 \uparrow 3 = 3^3 = 27$$

With two arrows it is an iterative exponentiation (double)

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

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

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

With three arrows

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

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