So can we say that in these examples, we are thinking of the imaginary unit i as a 90-degree rotation? But I am still kind of confused in terms of why that makes sense. How does the geometric understanding we are having here correspond to the algebraic meaning for imaginary number?

alyssal

|z1z2| = |z1||z2| = (a^2 + b^2)(c^2 + d^2), and the angle of z1z2 = theta1 + theta2 = atan((ad + bc)/(ac - bd)) using atan addition. The previous slide suggested that the angle addition was not quite correct, is that because it would have to be between -pi/2 to pi/2? I tried reading this stackexchange to understand: https://math.stackexchange.com/questions/326334/a-question-about-the-arctangent-addition-formula

So can we say that in these examples, we are thinking of the imaginary unit i as a 90-degree rotation? But I am still kind of confused in terms of why that makes sense. How does the geometric understanding we are having here correspond to the algebraic meaning for imaginary number?

|z1z2| = |z1||z2| = (a^2 + b^2)(c^2 + d^2), and the angle of z1z2 = theta1 + theta2 = atan((ad + bc)/(ac - bd)) using atan addition. The previous slide suggested that the angle addition was not quite correct, is that because it would have to be between -pi/2 to pi/2? I tried reading this stackexchange to understand: https://math.stackexchange.com/questions/326334/a-question-about-the-arctangent-addition-formula