On Odd-Rational Meadow Powers

       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.