Theorems – Reciprocals and Multiplicative Inverses

The theorems of reciprocals and 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. In R1 and R2: 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. In R3: a?^?-1 = 1‡a^‡a/a For example:
   • 1^{^-1} = 1^{^}/1 = 1^{^}
   • 1^v-1 = 1^v/1 = 1^v
   • i^{^-1} = i^{^}/1 = i^{^}
   • j^v-1 = j^v/1 = j^v
5. (a^?-1)^-1 = a if a is nonzero.  
   • For example R2:
     • (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
   • For example R3:
     • (9^{^-1})^-1 = (1^{^}/9)^-1  = (1/9^{^})^-1 =  1^{^}/(1/9) = 9^{^}
     • (9^v-1)^-1 = (1^v/9)^-1   = (1/9^v)^-1  =1^v/(1/9) = 9^v
     • (9i^{^-1})^-1 = (i^{^}/9)^-1  = (1/9i^{^})^-1 = i^{^}/(1/9) = 9i^{^}
     • (9i^v-1)^-1 = (i^v/9)^-1  = (1/9i^v)^-1 =   i^v/(1/9) = 9i^v
6. |1?^?-1| = 1 and is a Counter that cannot be used standalone
7. (^–a)^-1 = ^–(a^-1) if a is nonzero. 
   • For example R2: 
     • (^–9i^{^})^-1 = (9i^v)^-1 = (1^{^}/9i^v) = 1/9i^{^} and
     • ^–(9i^{^-1}) = ^–(1^{^}/9i^{^}) = ^–(1/9i^v) = 1/9i^{^} 
   • For example R3: 
     • (^–9i^{^})^-1 = (9i^v)^-1 = (i^v/9) = 1/9i^v and
     • ^–(9i^{^-1}) = ^–(i^{^}/9) = ^–(1/9i^{^}) = 1/9i^v
8. (ab)^-1 = a^-1b^-1 if a and b are nonzero.  
   • For example in R2:
     • (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
   • For example in R3:
     • (3i^v*2^v)^-1 = 6j^v-1 = j^v/6 = 1/6j^v
     • 3i^v-1*2^v-1 = (i^v/3)*(1^v/2) = 1/3i^v*1/2^v = 1/6j^v
9. (a/b)^-1 =  b/a if a and b are nonzero in R1 and R2.
   • For example in R2:
     • (6i^v/2^v)^-1 = 3i^{^-1} = 1^{^}/3i^{^} = 1/3i^v
     • 2^v/6i^v = 1/3i^v
   • It is not true in R3. For example:
     • (6i^v/2^v)^-1 = 3j^v-1 = j^v/3 = 1/3j^v
     • 2^v/6i^v = 1/3j^{^}

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