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Although I like the complex numbers and two dimensional real numbers being compared and contrasted (yes, R^2 with vector multiplication and and complex number c
by itchyjunk 5y ago
Although I like the complex numbers and two dimensional real numbers being compared and contrasted (yes, R^2 with vector multiplication and and complex number can be thought of as representing the same thing), I think this way of thinking misses that we also have a field of complex numbers. Point being R^2 isn't a field but C is.
If i remember, you don't get any other fields past this. No R^3,..,R^n. I also think this search for fields led to discovery of quaternion which doesn't have commutativity but is very close to being a field. So I find C as a field to be special (if not a useful distinction).
- asxd 5y ago"Field" is a term I've heard come up again and again since college, in engineering-adjacent math. Over the years, I've occasionally looked it up and tried to understand the importance, but I've never found any literature that made much sense to my admittedly short-sighted mind. I'm putting this bluntly (and sincerely), but this seems like a decent time to ask a question I should have asked long ago during college: What is a field and why do we care?
- thrtythreeforty 5y agoTo use C++ terminology, It's any type where operator+(), operator-(), operator*(), and operator/() are defined and behave as they do in rational numbers.