Negative Mathematics and Mathematical Symmetry

In my admittedly limited formal mathematical education, I was puzzled by two things:

1) Why, unlike addition and multiplication, was exponentiation not commutative?

2) Why, unlike for positive real numbers, is it much more difficult to do anything related to exponentiation with the negative real numbers?

The second question has always left me with the feeling that mathematics has a "right-handed" (positive) bias.

In hindsight, both are rather simple to address. Exponentiation can easily be made commutative as long as a positive number other than the multiplicative identity (in normal multiplication, 1) is chosen to be the identity element for exponentiation (see Abelian Exponentiation -- http://mysite.verizon.net/reriker/abex.html ).

The second I have addressed in a very general sense in the Mathematical Relativitiy papers (see http://mysite.verizon.net/reriker/re02000.html and http://mysite.verizon.net/reriker/re03000.html ) . However, I would like to address it specifically here. Again, it involves simply using a different identity element beginning with multiplication.

Negative Multiplication

In normal multiplication, the multiplicative identity is 1. Consider an alternative multiplication, N* (negative multiplication), where the multiplicative identity is -1 (the additive inverse of 1). Then,

a N* -1 = a

and

a N* b = -(a*b)

For example:

-1 N* -1 = -1
-1 N* -2 = -2
-2 N* -2 = -4
-1 N* -3 = -3
-2 N* -3 = -6
-3 N* -3 = -9
-1 N* -4 = -4
-2 N* -4 = -8
-3 N* -4 = -12
-4 N* -4 = -16
-1 N* -5 = -5
-2 N* -5 = -10
-3 N* -5 = -15
-4 N* -5 = -20
-5 N* -5 = -25
etc.


Negative Exponentiation

Normal exponentiation is a repeated number of multiplications of the same number (e.g., 2*2=2^2, 2*2*2=2^3, 2*2*2*2=2^4). Negative exponentiation (N^) is a repeated number of negative multiplications of the same number (e.g., -2 N* -2 = -2 ^N -2, -2 N* -2 N* -2 = -2 N^ -3, -2 N* -2 N* -2 N* -2 = -2 N^ -4).

For example:

-1 N^ -1 = -1
-2 N^ -1 = -2
-2 N^ -2 = -4
-2 N^ -3 = -8
-2 N^ -4 = -16
-3 N^ -1 = -3
-3 N^ -2 = -9
-3 N^ -3 = -27
-3 N^ -4 = -81
-4 N^ -1 = -4
-4 N^ -2 = -16
-4 N^ -3 = -64
-4 N^ -4 = -256
etc.


Negative Logarithms

If a,b are negative real numbers and

a N^ b = c

then

Nloga c = b

For example:

Nlog-2 -4 = -2
Nlog-2 -8 = -3
Nlog-3 -9 = -2
Nlog-3 -27 = -3
Nlog-4 -16 = -2
Nlog-4 -64 = -3
etc.


The Negative Primes

The prime numbers are the set of positive integers whose only factors are 1 and itself (i.e., 2,3,5,7,11,etc.). Consequently, the negative prime numbers are the set of negative integers whose only factors (with respect to N*) are -1 and itself (i.e.,-2,-3,-5,-7,-11,etc.).


The Boolean Logic of Negative Mathematics

~0=-1
~-1=0

0 OR 0 = 0
0 OR -1 = -1
-1 OR 0 = -1
-1 OR -1 = -1

0 AND 0 = 0
0 AND -1 = 0
-1 AND 0 = 0
-1 AND -1 = -1


Conclusion

As the Mathematical Relativity papers indicate, the field (R,+,N*,0,-1) is isomorphic to the normal Real field (R,+,*,0,1). Consequently, as these few items above begin to show, everything that has been shown to be true in the field (R,+,*,0,1) can be shown to have a valid counterpart in (R,+,N*,0,-1). Perhaps, as I have been repeatedly reminded, this isomorphism is trivial. However, I believe that the symmetry it restores to mathematics is not.