Theorems – Distributive

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

1. ^–(a + b) = (^–a) + (^–b) = ^–a + ^–b
2. ^–(a + ^–b) = ^–a + b
3. ^–(^–a + ^–b) = a + b
4. a + a = 2a
5. In R1 and R2:
   • a(b − c) = a*b − a*c = (b − c)a
   • Note that a(b − c) is the equivalent of a*(b − c) and (b − c)a the equivalent of (b − c)*a as multiplication with round brackets is defined earlier.
6. In R3:
   • a(b − c) = a*b−a*c ≠ (b − c)a. For example:
     • 3j^{^}(3^v + 2i^v) = 9i^v + 6^{^} ≠ (3^v + 2i^v)3j^{^} = 9i^{^} + 6^v
7. (a + b)(c + d) = a*c + a*d + b*c + b*d
8. In R1 and R2: (a + b)(c+^–d) = ac + a^–d + bc + b^–d = (c + ^–d)(a + b)
9. In R3: (a + b)(c+^–d) = ac + a^–d + bc + b^–d ≠ (c + ^–d)(a + b). For example:
   • (3^v + 2i^v)(3i^v + ^–2j^{^}) = 3^v*3i^v + 3^v*^–2j^{^} + 2i^v*3i^v + 2i^v*^–2j^{^}
     • = 9j^ + 3^v*2j^v + 6i^v + 2i^v*2j^v
     • = 9j^ + 6i^v + 6i^v + 4^{^}
     • = 4^ + 12iv + 9j^
   • (3i^v + ^–2j^{^})(3^v + 2i^v) = 4v + 9jv
10. (a + ^–b)(c + ^–d) = ac + a^–d + ^–bc + bd. For example, in R3:
    • (3^v + ^–2i^v)(3i^v + ^–2j^{^})
      • = 3^v*3i^v + 3^v*^–2j^{^} + ^–2i^v*3i^v + ^–2i^v*^–2j^{^}
      • = 9j^ + 3^v*2j^v + 2i^{^}*3i^v + 2i^{^}*2j^v
      • = 9j^ + 6i^v + 6i^v + 4^v
      • = 4^v + 12i^v + 9j^

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