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It's short, but I don't think that it's very helpful for someone who hears about it for the first time. When you see a + bi your first reaction, as a 12 year
by golergka 6y ago
It's short, but I don't think that it's very helpful for someone who hears about it for the first time. When you see
a + bi
your first reaction, as a 12 year old, is to look at the plus sign and wonder - what's the actual result here is? How do these things add up?
I think that explaining it through geometry and vectors first is far more natural to begin with, and gives a solid intuition of what happens with complex numbers when they go through simpler operations.
- Koshkin 6y agoThis 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.