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

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Pro<strong>of</strong>.
Rational numbers a; b 2 Q take the form a=([i]) and b=([k]);
where i; k; j; l 2 I with j; l 6= . The product <strong>of</strong> these rationals is also in Q because
the product <strong>of</strong> a and b reduces to the rational ([m]); where m=([i][k]) and
n=([j][l]):
([ a ][ b ]) Given
([([i])][([k])]) Replacement
( [i] [k] ) Involution
( [i] [k] ) Inverse collection
([([i] [k])]) Involution
([ m ]) Replacement
Since the product <strong>of</strong> two integers is an integer, both m and n are integers. Because
j and l are non-void, n is also non-void. Therefore, ([m]) is rational and so
the product <strong>of</strong> a and b is rational.
Every element inQ, except for the void, has a multiplicative inverse.
Pro<strong>of</strong>. Rational number a 2 Q takes the form a=([i]); where i; k 2 I with
j 6= . Since a is non-void, the numerator is also non-void, i 6= . The multiplicative
inverse <strong>of</strong> a, given by (); is rational by the following reduction back to the
rational form.
()=()=()=([j])
5.4.3 Division by Zero
Dominion, a2=2, loses information and necessitates excluding from the calculus.
This restriction prevents formation <strong>of</strong> division by zero, (): This restriction
appears in standard and boundary numbers as:
1=0 undened versus undened.
5.4.4 Calculation
Computing a division requires a matching <strong>of</strong> quantities. When quantities cannot be
matched, the fraction does not reduce. Thus, the calculus constrains this calculation
but does not guide the reduction.