21–30 of 143 posts
The title statement is meaningless. "Imaginary numbers" are not "multiply" by "90°". What does it even mean to "multiply" by an angle? Correct statement: "*i is equivalent to a 90° rotation in some contexts." Which is kind of obvious to anybody who has done some maths or physics at some point?
Why does (-1)*(-1)=(+1) ? It is arbitrary, and there really is no good reason. We could construct number lines that work differently, so that imaginary numbers never appear. Such alternative number lines would still allow us to solve the exact same physics and engineering problems. Sure, the computations would work differently, but the way we would measure and use the initial conditions in our equations would be diff…
Why does (-1)*(-1)=(+1) ? It is arbitrary, and there really is no good reason. We could construct number lines that work differently, so that imaginary numbers never appear. Such alternative number lines would still allow us to solve the exact same physics and engineering problems. Sure, the computations would work differently, but the way we would measure and use the initial conditions in our equations would be diff…
Maybe I missed something in the explanation. Can somebody explain?
I don't think I quite followed the step where he/she writes "i^4 = 1", where they are relating rotation and the natural numbers. The RHS is theoretically the concept "identity under rotation". Why should it be be the same as the natural number 1? Maybe I missed something in the explanation. Can somebody explain?
I don't think I quite followed the step where he/she writes "i^4 = 1", where they are relating rotation and the natural numbers. The RHS is theoretically the concept "identity under rotation". Why should it be be the same as the natural number 1? Maybe I missed something in the explanation. Can somebody explain?
* i * i = i^2 = sqrt(-1) * sqrt(-1) = -1
..
i^4 = i^2 * i^2 = -1 * -1 = 1
Why does (-1)*(-1)=(+1) ? It is arbitrary, and there really is no good reason. We could construct number lines that work differently, so that imaginary numbers never appear. Such alternative number lines would still allow us to solve the exact same physics and engineering problems. Sure, the computations would work differently, but the way we would measure and use the initial conditions in our equations would be diff…
Why does (-1)*(-1)=(+1) ? It is arbitrary, and there really is no good reason. We could construct number lines that work differently, so that imaginary numbers never appear. Such alternative number lines would still allow us to solve the exact same physics and engineering problems. Sure, the computations would work differently, but the way we would measure and use the initial conditions in our equations would be diff…
It's not arbitrary. It comes directly from the axiomatics.