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When I was learning complex numbers, the teacher didn't want us to assume sqrt(-1) = i, we had to go like this: sqrt(-1) = sqrt(e^Pi) = e^(Pi/2) = i So the ne
by matstc 18y ago
When I was learning complex numbers, the teacher didn't want us to assume sqrt(-1) = i, we had to go like this:
sqrt(-1) = sqrt(e^Pi) = e^(Pi/2) = i
So the new concept was really a 2D plane for numbers, not a new definition for sqrt of negative integers.
- antiform 18y agoActually, even the definition of this form requires i, because it's e^(i)(pi/2) = i. The simple assumption that sqrt(-1) = i is problematic because you can get something like i^2 = (sqrt(-1))^2 = sqrt((-1)^2) = 1. The wonder and surprise of complex numbers is that assuming seemingly arbitrary properties of a constant like i lead to a number of deep and beautiful results.