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If you give vectors in the plane the same multiplication rule as complex numbers, then I think it's fair to say that they are ontologically equivalent. If you t
by joe__f 5y ago
If you give vectors in the plane the same multiplication rule as complex numbers, then I think it's fair to say that they are ontologically equivalent. If you take your vectors and equip them with a different multiplication rule they will no longer form a field. Any other structure that your vectors may form under this multiplication rule is then ontologically different from the complex numbers.
Wikipedia tells me that "Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra." The fact that the complex numbers can form the scalars of a vector space of all possible quantum states is an essential property for formulating quantum mechanics.
I think your point that the complex numbers don't have anything unreal about them is probably fair. One key difference between the real and complex numbers is that the complex numbers have no canonical ordering, so you can't use them to measure distances between points in physical space. Maybe in this sense you could think of them as being less real than the reals or integers etc.
- lokedhs 5y agoI don't fully understand what you meant. Either I misunderstood it or there is a problem with your explanation. You can measure distance between complex numbers. But the distances between three numbers has a non-transitive relationship which is what prevents ordering. Was that what you meant, or is there some more fundamental relation between not having an order and the idea of distances?
- joe__f 5y agoReal numbers have a unique canonical ordering (up to isomorphism). So given x and y real, it is always true that either x < y, x = y or x > y. Complex numbers don't have a unique ordering. So for for example take 1+i and 5-2i. Which is bigger? There are many possible orderings that you can define on the complex numbers. The fact that ordering them is not unique means that any such order is a choice rather than a property of the numbers themselves. You're thinking about measuring distances in the complex plane which is absolutely fine, and also has a canonical definition. I was referring to the fact that it's not intrinsically meaningful to say that 'London is (5000+250i)km from New York'
- galaxyLogic 5y agoComplex numbers are strange and wonderous because of the rules about their interactions including how they always seem to be a sum of an imaginary and real part.