Theorems – Multiplicative Inverses

The theorems of multiplicative inverses are derived from the theorems in Axioms for Real Numbers as interpreted for Wave Numbers.

1. Multiplicative inverses exist in R1 and R2 but not in R3
2. If a is nonzero, then so is a^-1
3. a?^?-1 = 1^{^}/a‡a^‡a For example:
   • 1^{^-1} = 1^{^}/1^{^} = 1^{^}
   • 1^v-1 = 1^{^}/1^v = 1^v
   • i^{^-1} = 1^{^}/i^{^} = i^v
   • i^v-1 = 1^{^}/i^v = i^{^}
   • 4^v-1 = 1^{^}/4^v = 0.25^v
   • 5i^{^-1} = 1^{^}/5i^{^} = 0.2i^v
4. (a^?-1)^-1 = a if a is nonzero.  For example:
   • (9^{^-1})^-1 = (1^{^}/9^{^})^-1  = (1/9^{^})^-1 =  1^{^}/(1/9^{^}) = 9^{^}
   • (9^v-1)^-1 = (1^{^}/9^v)^-1   = (1/9^v)^-1  =1^{^}/(1/9^v) = 9^v
   • (9i^{^-1})^-1 = (1^{^}/9i^{^})^-1  = (1/9i^v)^-1 = 1^{^}/(1/9i^v) = 9i^{^}
   • (9i^v-1)^-1 = (1^{^}/9i^v)^-1  = (1/9i^{^})^-1 =   1^{^}/(1/9i^{^}) = 9i^v
5. |1?^?-1| = 1 and is a Counter that cannot be used standalone
6. (^–a)^-1 = ^–(a^-1) if a is nonzero.  For example: 
   • (^–9i^{^})^-1 = (9i^v)^-1 = (1^{^}/9i^v) = 1/9i^{^} and
   • ^–(9i^{^-1}) = ^–(1^{^}/9i^{^}) = ^–(1/9i^v) = 1/9i^{^} 
7. (ab)^-1 = a^-1b^-1 if a and b are nonzero.  For example:
   • (3i^v*2^v)^-1 = 6i^{^-1} = 1^{^}/6i^{^} = 1/6i^v
   • 3i^v-1*2^v-1 = (1^{^}/3i^v)*(1^{^}/2^v) = 1/3i^{^}*1/2^v = 1/6i^v
8. (a/b)^-1 =  b/a if a and b are nonzero. For example:
   • (6i^v/2^v)^-1 = 3i^{^-1} = 1^{^}/3i^{^} = 1/3i^v
   • 2^v/6i^v = 1/3i^v

Next: Quotients

Previous: Distributive