Theorems – Inequalities

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

Transitivity

1. if a < b and b < c, then a < c
2. If a ≤ b and b < c, then a < c
3. If a < b and b ≤ c, then a < c
4. If a ≤ b and b ≤ c, then a ≤ c

Other Properties

1. If a is non-zero, 0 < a
2. If a ≤ b and b ≤ a, then |a| = |b|.
3. If a < b, then ^–a < ^–b.
4. 0 < ?^{^};                   0 < ?^v
5. If a is non-zero and a < b, then a^-1 > b^-1
6. If a < b and c < d, then a + c < b + d
7. If a ≤ b and c < d, then a +c < b + d
8. If a ≤ b and c ≤ d, then a + c ≤ b + d
9. If c is non-zero and a < b, then ac < bc
10. If a < b and c < 0 is not possible – no Opposite Value < 0
11. If a ≤ b, then ac ≤ bc
12. If a < b and c < d, then ac < bd
13. If a ≤ b and c ≤ d, then ac ≤ bd
14. ab > 0 if neither a nor b = 0
15. ab < 0 is not possible
16. There does not exist a smallest ^{^} Opposite Value. There does not exist a smallest ^v Opposite Value. The smallest Opposite Value is 0.
17. (Density) If a and b are two distinct Opposite Values, then there exist infinitely many rational numbers and infinitely many irrational numbers between a and b

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