Is it possible to determine if one complex number is greater than another? Or as the question implies is there an "order" to complex numbers (like 1 is before 2 in the real numbers)?
I thought that would could simply use the modulus to determine if one complex numbers is greater than another, though I believe this can't be the only way used (what if 2 complex numbers have the same modulus and are quite different). So I thought, if the point is in the uppermost right quadrant of the complex plane, then both real and imaginary parts are positive, so it would be greater than any other complex number in a different quadrant of the complex plane (you might say what about the modulus, but in the reals $1>-2$ even though $|-2|>|1|$). But what if one complex number has a positive real, and negative imaginary and another one has a negative real and positive imaginary? (And for arguments sake they both have the same modulus)
If we can't determine why not? In the real numbers it seems (to me), quit trivial at a basic level to determine if one real is greater than another e.g. $2>1$. What is this property of numbers called? Why doesn't complex numbers exhibit this property (if indeed it doesn't)?
Answer
It is possible to order the complex numbers. For instance, one could define $x_1+iy_1 However, it's impossible to define a total order on the complex numbers in such a way that it becomes an ordered field. This is because in an ordered field the square of any non-zero number is $>0$. Hence we would have $-1=i^2>0$, and adding $1$ to both sides would imply $0>1=1^2$, which is a contradiction.
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