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)
Famous quotes containing the words complex, numbers and/or inequalities:
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“The only inequalities that matter begin in the mind. It is not income levels but differences in mental equipment that keep people apart, breed feelings of inferiority.”
—Jacquetta Hawkes (b. 1910)