Theorems – Quotients

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

1. Division by 0 is undefined
2. Division is not associative
3. a/1 = a
4. R1 and R2: 1^{^}/a = a^-1 if a is nonzero
5. R3: 1‡a^‡a/a = a^-1 if a is nonzero
6. R1 and R2: a/a = 1^{^} if a is nonzero
7. R3: a?^?/a‡a^{^} = a?^?/a‡a^v= 1‡a^‡a if a is nonzero
8. (a/b)(c/d) = (a*c)/(b*d) if b and d are nonzero
   • R2: (8i^v/4^v)(12i^{^}/4^{^}) = 2i^{^}*3i^{^} = 6^v = (8i^v*12i^{^})/(4^v*4^{^}) = 96^{^}/16^v = 6^v
   • R3: It is not always true as multiplication and division are not associative
     • (8j^v/4i^v)(12i^{^}/4^v) = 2^v*3j^{^} = 6i^{^} = (8j^v*12i^{^})/(4i^v*4^v) = 96^{^}/16j^v = 6i^{^}, however
     • (8^v/4i^v)*(12j^{^}/4^v) = 2j^{^}*3i^v = 6^{^} ≠ (8^v*12j^{^})/(4i^v*4^v) = 96i^{^}/16j^v = 6^v
9. (a/b)/(c/d)  = (a*d)/(b*c) if b, c and d are non-zero
   • R2: (8i^v/4^v)/(12i^{^}/4^{^}) = 2i^{^}/3i^{^} = 2/3^{^} = (8i^v*4^{^})/(4^v*12i^{^}) = 32i^v/48i^v = 2/3^{^}
   • R3: It is not always true as multiplication and division are not associative
     • (8i^v/4j^v)/(12i^{^}/4j^{^}) = 2^{^}/3^{^} = 2/3^{^} ≠ (8i^v*4j^{^})/(4j^v*12i^{^}) = 32^v/48^{^} = 2/3^v.
10. (a/b)/(c/d)  = (d*a)/(b*c) if b, c and d are nonzero:
    • R2: (8i^v/4^v)/(12i^{^}/4^{^}) = 2i^{^}/3i^{^} = 2/3^{^} = (4^{^}*8i^v)/(4^v*12i^{^}) = 32i^v/48i^v = 2/3^{^}
    • R3: It is not always true as multiplication and division are not associative
      • (8i^v/4j^v)/(12i^{^}/4j^{^}) = 2^{^}/3^{^} = 2/3^{^} = (4j^{^}*8i^v)/(4j^v*12i^{^}) = 32^{^}/48^{^} = 2/3^{^}
      • (8i^v/4i^v)/(12i^{^}/4j^{^}) = 2i^v/3^{^} = 2/3j^{^} = (4j^{^}*8i^v)/(4i^v*12i^{^}) = 32^{^}/48i^{^} = 2/3j^{^}, however
      • (6^v/2^{^})/(4i^v/2^{^}) = 3^v/2j^{^} = 3/2i^{^} ≠ (2^{^}*6^v)/(4i^v*2^{^}) = 12^v/8j^{^} = 3/2i^{^}
11. (a*c)/(b*c) =  a/b if b and c are nonzero
    • R2: (8i^v*4^v)/(12i^{^}*4^v) = 32i^{^}/48i^v = 2/3^v = 8i^v/12i^{^} = 2/3^v
    • R3: It is not always true as multiplication and division are not associative
      • (8i^v*4j^v)/(12^{^}*4j^v) = 32^{^}/48i^{^} = 2/3j^{^} = 8i^v/12^{^} = 2/3j^{^} however,
      • (6i^{^}*2j^v)/(4i^{^}*2j^v) = 12^v/8^v = 3/2^v ≠ 6i^{^}/4i^{^} = 3/2i^{^} and
      • (6^v*2^{^})/(4i^v*2^{^}) = 12^{^}/8j^{^} = 3/2i^v ≠ 6^v/4i^v = 3/2j^{^}
12. a(b/c) = (a*b)/c if c is nonzero
    • R2: (8i^v(4^v/2i^{^}) =  8i^v*2i^{^} = 16^{^}  =  (8i^v*4^v)/2i^{^} = 32i^{^}/2i^{^} = 16^{^}
    • R3: It is not always true as multiplication and division are not associative:
      • (8i^v(4j^v/2i^{^}) =  8i^v*2^{^} = 16j^{^} =  (8i^v*4j^v)/2i^{^} = 32^{^}/2i^{^} = 16j^{^}, however
      • 8j^{^}*(6i^v/3^{^}) = 8j^{^}*2j^{^} = 16j^{^} ≠ (8j^{^}*6i^v)/3^{^} = 48^{^}/3^{^} = 16^{^}
      • 8j^{^}*(6j^v/3i^{^}) = 8j^{^}*2^{^} = 16i^{^} ≠ (8j^{^}*6j^v)/3j^{^} = 48j^v/3j^{^} = 16j^v
13. (a*b)/b = a if b is nonzero
    • R2: (8i^v*4^v)/4^v =  32i^{^}/4^v = 8i^v 
    • R3: It is not always true as multiplication and division are not associative:
      • (8i^v*4j^v)/4j^v =  32^{^}/4j^v = 8i^{^} ≠ 8i^v
14. (^–a)/b = ^–(a/b) if b is nonzero
    • R2: (^–8i^v)/4^v = 8i^{^}/4^v = 2i^v =   ^–(8i^v/4^v) = ^–2i^{^} = 2i^v
    • R3: (^–8i^v)/4i^v = 8i^{^}/4i^v = 2i^{^} =   ^–(8i^v/4i^v) = ^–2i^v = 2i^{^}
15. (^–a)/(^–b) = a/b if b is nonzero
    • R2: (^–8i^v)/(^–4^v) = 8i^{^}/4^{^} = 2i^{^} =   8i^v/4^v = 2i^{^}
    • R3: (^–8i^v)/(^–4j^v) = 8i^{^}/4j^{^} = 2^{^} =   8i^v/4j^v = 2^{^}
16. a/b + c/d  = (a*d + b*c)/(b*d) if b and d are nonzero
    • R2: (8i^v/4^v) + (12i^{^}/4^{^}) = 2i^{^} + 3i^{^} = 5i^{^} = (8i^v*4^{^} + 4^v*12i^{^})/(4^v*4^{^}) = (32i^v + 48i^v )/(16^v) = 5i^{^}
    • R3: It is not always true as multiplication and division are not associative. For example:
      • (8i^v/4j^v) + (12i^{^}/4j^{^}) = 2^{^} + 3^{^} = 5^{^} ≠ (8i^v*4j^{^} + 12i^{^}*4j^v)/(4j^v*4j^{^}) = (32^v + 48^v)/(16^v) = 5^v
17. a/b + c/d  = (d*a + b*c)/(b*d)  if b and d are nonzero
    • R2: (8i^v/4^v) + (12i^{^}/4^{^}) = 2i^{^} + 3i^{^} = 5i^{^} = (4^{^}*8i^v + 4^v*12i^{^})/(4^v*4^{^}) = (32i^v + 48i^v )/(16^v) = 5i^{^}
    • R3: It is not always true as multiplication and division are not associative:
      • (8i^v/4j^v) + (12i^{^}/4j^{^}) = 2^{^} + 3^{^} = 5^{^} ≠ (4j^{^}*8i^v + 4j^v*12i^{^})/(4j^v*4j^{^}) = (32^{^} + 48^{^} )/(16j^{^}) = 5i^v
18. a/b + ^–c/d  = (a*d + b*^–c)/(b*d) if b and d are nonzero
    • R2: (8i^v/4^v) + (^–12i^{^}/4^{^}) = 2i^{^} + 3i^v = 1i^v = (8i^v*4^{^} + 4^v*^–12i^{^})/(4^v*4^{^}) = (32i^v + 48i^{^} )/(16^v) = 1i^v
    • R3. It is not always true as multiplication is not associative
      • (8i^v/4j^v) + (^–12i^{^}/4j^{^}) = 2^{^} + 3^v = 1^v ≠ (8i^v*4j^v + 4j^v*^–12i^{^})/(4j^v*4j^{^}) = (32^{^} + 48^v )/(16j^{^}) = i^{^}
19. a/b + ^–c/d  = (d*a + b*^–c)/(b*d) is true if b and d are nonzero
    • R2: (8i^v/4^v) + (^–12i^{^}/4^{^}) = 2i^{^} + 3i^v = 1i^v = (4^{^}*8i^v + 4^v*^–12i^{^})/(4^v*4^{^}) = (32i^v + 48i^{^} )/(16^v) = 1i^v
    • R3. It is not always true as multiplication is not associative
      • (8i^v/4j^v) + (^–12i^{^}/4j^{^}) = 2^{^} + 3^v = 1^v ≠ (4j^{^}*8i^v + 4j^v*^–12i^{^})/(4j^v*4j^{^}) = (32^{^} + 48^v )/(16j^{^}) = i^{^}
20. In R2 (a + bi)/(a + bi) = 1^{^}
21. If a/b = c, then b*c = a
    • R1:   
      • 6^{^}/3^v = 2^v, 3^v*2^v = 6^{^}
      • 6^{^}/6^{^} = 1^{^}, 6^{^}*1^{^} = 6^{^}
    • R2:               
      • (6^{^}+ 3i^v)/3^v = (2^v + 1i^{^}), 3^v(2^v + 1i^{^}) = (6^{^} + 3i^v)      
      • (6^{^}+ 3i^v)/(6^{^}+ 3i^v) = 1^{^},  (6^{^}+ 3i^v)*1^{^} = (6^{^}+ 3i^v)
    • R3:               
      • 6^{^}/3^v = 2^{^}, 3^v*2^{^} =  6^{^}
      • 6^{^}/6^{^} = 1^{^}, 6^{^}*1^{^} = 6^{^}
      • (2^{^} + 3i^v + 4j^{^})/(2^{^} + 3i^v + 4j^{^}) = (0.569^{^} + 1.033i^v + 1.089j^{^})
        • (2^{^} + 3i^v + 4j^{^})*(0.569^{^} + 1.033i^v + 1.089j^{^}) = (2^{^} + 3i^v + 4j^{^})

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