Complex Numbers and Inequalities
The set of complex numbers with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that becomes an ordered field. To make an ordered field, it would have to satisfy the following two properties:
- if a ≤ b then a + c ≤ b + c
- if 0 ≤ a and 0 ≤ b then 0 ≤ a b
Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ ). In either case 0 ≤ a2; this means that and ; so and, which means ; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:
- a ≤ b if < or ( and ≤ )
It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.
Read more about this topic: Inequality (mathematics)
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