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Proof

(related to Proposition: Complex Numbers Cannot Be Ordered)

We provide a proof by contradiction. * Assume, the field of complex numbers (\mathbb C,+,\cdot) is an ordered field, i.e. there is a linear order\geq “ on \mathbb C, which fulfills the two properties: * from z_1\geq z_1 it follows z_1+z_3\geq z_1+z_1 (for arbitrary z_1,z_2,z_3\in \mathbb C), and * from z_1\geq 0 and z_2\geq 0 it follows z_1\cdot z_2\geq 0 (for z_1,z_2\in F). * It suffices to show that the two complex numbers, the imaginary unit i\in\mathbb C and the complex zero 0\in\mathbb C, cannot be ordered to fulfill the second property. * Case i \ge 0. * Then i\cdot i\ge 0\cdot i=0, or -1 \ge 0 which is a contradiction in the embedded real numbers as a special case of complex numbers. * Case i < 0. * Then (-i)\cdot (-i) > 0, or -1 > 0 which is again a contradiction.


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