**On Odd-Rational Meadow Powers**

This
paper combines “number meadows”, in which negative numbers do not cancel in
fractions, and “odd rationals”, which defines triple addition but not double
addition; the resulting “odd-rational meadow” governs the one-to-one rational
power functions.

**Number Meadows**

A
number meadow is like an ordered number field, except that the cancellation law
applies only to positive numbers. In the rationals, the equation

(ac)
/ (bc) = a / b

applies
for any c not equal to zero; in a number meadow it applies only for positive
values of c. Therefore we get these new numbers:

0/-1 = @ = ‘additive
alternator’

-1/-1 = #1 = ‘multiplicative
alternator’, or ‘alternate positive’

1/-1 = {-}1 = ‘alternate
negative’.

The usual
algebraic rules imply these identities:

n/m +
0/-1 =
-n/-m; that is; X + @ = #X

@ +
0 =
@*0 = @ ; @
+ @ = @*@ = 0

##x =
x ;
#-x = -#x = {-}x ; #{-}x = {-}#x = -x ; {-}-x = -{-}x = #x

X +
#Y = #(X+Y) ; X * #Y = #(X*Y) ; 1/#X =
#(1/X); -#X=#-X

#X +
#Y = (X+Y) ; #X * #Y = (X*Y) ; #X/#Y
= X/Y

In a
number meadow there are four signs: positive (+/+), negative (-/+), alternate
positive (-/-), and alternate negative (+/-). These signs multiply in a Klein
4-group. Numbers with the first two signs are mainstream, numbers with the
second two are alternate. If you add,
subtract, multiply or divide a mainstream number with an alternate number, then
you get an alternate number; and alternate with alternate yields mainstream.

In a
number meadow, 0x equals 0 on the mainstream numbers, but @ on the alternates.
These laws apply:

0x *
0y = 0(xy) = 0x + 0y = 0(x+y)

-0x =
0x ;
#0x = {-}0x = @x

In a
number meadow, normal two-term distribution does not work:

@ *
(@+@) =
@*0 = @ ; but @*@+@*@
= 0+0 = 0

However,
three-term distribution does:

x*(a+b+c) =
xa+xb+xc for any
x, a, b, c.

Many of
the field laws still apply; commutativity, associativity, inverses. Division by
zero is still problematic; but now we can distinguish between positive and
negative infinity. In fact:

-1/0 =
1/@ = #(1/0)
= 1/0 + @

The
additive alternator @ adds like Z mod 2; the multiplicative alternator #1 multiplies
like Z mod 2; which suggests an isomorphism, such as exponentiation. Therefore I
propose the following definition for meadow exponentiation and logarithm. For
any finite nonzero number A;

A^@
= #1 ; log

_{A}(#1) = @
A^(@+x) = A^(#x) = @ +
A^x =
# A^x

log

_{A}(@+x) = log_{A}(#x) = @+log_{A}(x) = # log_{A}(x)**Odd Rational Meadow**

An

*odd rational*is a ratio n/m where both n and m are odd numbers. Therefore neither zero nor infinity are odd rationals.
The
odd rationals are closed under multiplication:

(odd/odd)*(odd/odd) =
odd/odd

They
are not closed under addition:

(odd/odd)+(odd/odd) = even/odd

But
they are closed under triple-addition:

(odd/odd)+(odd/odd)
+(odd/odd) = odd/odd

In
the odd rationals, multiplication, triple addition, negatives and reciprocals
are defined everywhere. All the usual algebraic laws apply, though distribution
is necessarily triple distribution, and since zero is not an odd rational, we
must replace the identity and inverses law with the cancellation law x+y-y=x.

To
the odd rationals add the alternator @ and the alternate signs # and {-}; deny
cancellation of negatives and you get the

*odd rational meadow.*It has the laws of a number meadow, as noted in the previous section.**Odd Rational Meadow Powers**

Consider
the rational power functions on the real numbers; F(x) = x^(n/m). Such functions are defined at -1
only if m is odd; they are one-to-one and onto only if n is odd. Therefore
rational power functions are complete, one-to-one and onto if and only if n/m
is an odd rational.

We
compose power functions by multiplying powers:

(X^(a/b))^(c/d) = X ^
((a/b)*(c/d))

If we
double-multiply power functions then the result is neither one-to-one nor onto.
So one-to-one, complete, onto odd-rational power functions are closed under
triple but not double multiplication.

We
multiply triples of power functions by triple-adding powers:

(X^(a/b))*(X^(c/d))*(X^(e/f)) = X ^
((a/b)+(c/d)+(e/f))

Alternators
in powers give access to alternate numbers:

X^(a+@) = @
+ X^a

This
implies a discontinuity at infinity:

2^(1/0) =
1/0 ; but 2^(@+1/0)
= 2^(-1/0) = 0, not equal to @+1/0.

So we
can compose, alternate and triple-multiply power functions with odd rational
meadow powers.

## No comments:

## Post a Comment