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This is perhaps true for the complex numbers, but I am not sure if any field extension can be explained geometrically in a sensible way. (For example, think abo
by Koshkin 6y ago
This is perhaps true for the complex numbers, but I am not sure if any field extension can be explained geometrically in a sensible way. (For example, think about extending the rationals by adding the square root of 2.)
- jacobolus 6y agoAdding the square root of 2 gives you a way of describing coordinates in a symmetry system where you start with a square grid and then allow yourself to make 1/8 turn rotations.
- golergka 6y agoThat's why you first develop a very good intuition of what complex numbers really are, understand Euler's formula (and probably asked to come up with a proof for it), and only then move forward. In mathematics, it's really easy and dangerous to think that you understand a concept only because you can memorize it's definitions and properties; any math education should fight this pseudo-understanding and make sure that you really grok it.
- Koshkin 6y agoWhat you are saying is true of course, except nobody in their right mind should think that algebra lacks intuition and that learning it consists of rote memorization.