Inequality (mathematics) - Complex Numbers and Inequalities

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 ab then a + cb + 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 ab then a + cb + c"). Sometimes the lexicographical order definition is used:

  • a ≤ b if < or ( and ≤ )

It can easily be proven that for this definition ab implies a + cb + c.

Read more about this topic:  Inequality (mathematics)

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