A Calculus of Number Based on Spatial Forms - University of ...

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Multiplication in the calculus can be seen as an adding <strong>of</strong> logarithms:
xy = e ln(xy) = e ln x+ln y ! ([x][y])
Exponentiation in the calculus can be seen as a multiplication <strong>of</strong> a logarithm:
x y = e ln xy = e y ln x = e eln(y ln x) = e eln y+ln ln x ! (([[x]][y]))
The logarithmic formulas connect multiplicative operations with additive operations.
Because the forms <strong>of</strong> calculus can be interpreted as logarithms, these transformations
are trivial.
ln(xy) =lnx+lny versus [([x][y])]=[x][y]
ln x=y =lnx,ln y versus [([x])]=[x]
ln x y = y ln x versus [(([[x]][y]))]=([[x]][y])
6.5 Transcendentals
Transcendental numbers, such as Euler's number and an algebraic , can be represented
in the calculus as can basic trigonometric functions.
6.5.1 Euler's <strong>Number</strong>, e
The instance boundary acts as an exponential. A double nesting <strong>of</strong> instance equals
its base, e e0 = e ! (): Setting the base <strong>of</strong> the function to Euler's number makes this
boundary form represent Euler's number. This nesting does not reduce to any other
boundary form, making the transcendental value incommensurable with previously
dened numbers.
6.5.2 Pi,
Interpreting the abstract boundary as the natural logarithm makes J =ln,1=i.
This interpretation fullls Euler's formula translated to J wrapped by instance:
1+e i =0 versus ([])=