- #1
jgens
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Would the following prove that the set of complex numbers do not form and ordered field?
Clearly [itex]i \neq 0[/itex]. Therefore, if the complex numbers form an ordered field either [itex]i > 0[/itex] or [itex]i < 0[/itex]. Suppose first that [itex]i > 0[/itex], then [itex]i^2 = -1 > 0[/itex], a contradiction. Now suppose that [itex]i < 0[/itex], then [itex]i^2 = -1 > 0[/itex], another contradiction. Thus, the set of complex numbers do not form an ordered field.
This seems awfully fishy and I wouldn't be surprised to find that it's completely invalid. Feedback and suggestions are welcome. Thanks!
Clearly [itex]i \neq 0[/itex]. Therefore, if the complex numbers form an ordered field either [itex]i > 0[/itex] or [itex]i < 0[/itex]. Suppose first that [itex]i > 0[/itex], then [itex]i^2 = -1 > 0[/itex], a contradiction. Now suppose that [itex]i < 0[/itex], then [itex]i^2 = -1 > 0[/itex], another contradiction. Thus, the set of complex numbers do not form an ordered field.
This seems awfully fishy and I wouldn't be surprised to find that it's completely invalid. Feedback and suggestions are welcome. Thanks!