The theorems of multiplicative inverses are derived from the theorems in Axioms for Real Numbers as interpreted for Wave Numbers.
- Multiplicative inverses exist in R1 and R2 but not in R3
- If a is nonzero, then so is a-1
- a??-1 = 1^/a‡a‡a For example:
- 1^-1 = 1^/1^ = 1^
- 1v-1 = 1^/1v = 1v
- i^-1 = 1^/i^ = iv
- iv-1 = 1^/iv = i^
- 4v-1 = 1^/4v = 0.25v
- 5i^-1 = 1^/5i^ = 0.2iv
- (a?-1)-1 = a if a is nonzero. For example:
- (9^-1)-1 = (1^/9^)-1 = (1/9^)-1 = 1^/(1/9^) = 9^
- (9v-1)-1 = (1^/9v)-1 = (1/9v)-1 =1^/(1/9v) = 9v
- (9i^-1)-1 = (1^/9i^)-1 = (1/9iv)-1 = 1^/(1/9iv) = 9i^
- (9iv-1)-1 = (1^/9iv)-1 = (1/9i^)-1 = 1^/(1/9i^) = 9iv
- |1??-1| = 1 and is a Counter that cannot be used standalone
- (–a)-1 = –(a-1) if a is nonzero. For example:
- (–9i^)-1 = (9iv)-1 = (1^/9iv) = 1/9i^ and
- –(9i^-1) = –(1^/9i^) = –(1/9iv) = 1/9i^
- (ab)-1 = a-1b-1 if a and b are nonzero. For example:
- (3iv*2v)-1 = 6i^-1 = 1^/6i^ = 1/6iv
- 3iv-1*2v-1 = (1^/3iv)*(1^/2v) = 1/3i^*1/2v = 1/6iv
- (a/b)-1 = b/a if a and b are nonzero. For example:
- (6iv/2v)-1 = 3i^-1 = 1^/3i^ = 1/3iv
- 2v/6iv = 1/3iv
Next: Quotients
Previous: Distributive