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As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we lose moving from the reals to the complex numbers?

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    Possible duplicate of [What is lost when we move from reals to complex numbers?](https://math.stackexchange.com/questions/2728317/what-is-lost-when-we-move-from-reals-to-complex-numbers)2018-04-09

2 Answers 2

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The most obvious property that we lose is the linear (or total) ordering of the real line.

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    You cannot use <, >, ≤, ≥ in a way which is backward compatible with how these are defined for the reals, yet makes sense in the complex plane.2012-11-20
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We lose equality of the complex conjugate and total order.

So: $x+i y \ne x-i y$ for complex numbers which are not also reals.

And you can't say wether $ i > -i $ or $i < -i$, etc.
All you have is the magnitude which, in the given example, is equal.

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    Magnitude does not map into the reals in a way that is "backward compatible" with how reals are compared; i.e. that -2 is less than 1.2012-11-20