Theorems – Flipping

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

1. ^–0 = 0.
2. ^–(^–a) = a.
3. R1 and R2: ^–a*b = ^–(a*b)  For example:
   • ^–1^{^}*1^v = 1^v*1^v = 1^{^} = ^–(1^{^}*1^v) = ^–(1^v) = 1^{^}
   • ^–i^{^}*1^v = i^v*1^v = i^{^} = ^–(i^{^}*1^v) = ^–(i^v) = i^{^}
4. R3 and higher: ^–a*b ≠ ^–(a*b)  For example:
   • ^–j^{^}*j^v = j^v*j^v = j^v ≠ ^– (j^{^}*j^v) = ^–(j^v ) = j^{^}
5. R1 and R2: ^–a*^–b = a*b  For example:
   • ^–1^{^}*^–1^v = 1^v*1^{^} = 1^v = 1^{^}*1^v = 1^v
   • ^–i^{^}*^–1^v = i^v*1^{^} = i^v = i^{^}*1^v = i^v
6. R3 and higher: ^–a*^–b ≠ a*b  For example:   
   • ^–j^{^}*^–j^v  = j^v*j^{^} = j^{^} ≠  j^{^}*j^v = j^v
7. ^–a?^? = |a|*^–1‡a^‡a
8. ^–a?^{^}  = a?^v;           ^–a?^v  = a?^{^};

Next: Distributive

Previous: Zero