A calculus of the absurd

15.2.2 Loci on the Argand diagram

Consider the expression $$\abs {z - i} = 1$$. One way to think about this is using vectors. Here $$z$$ is a variable, standing for any vector in the complex plane. Read aloud, this equation means something along the lines of “the distance between $$i$$ and $$z$$ is equal to $$1$$”. Why the distance? Because $$z$$ and $$i$$ can be treated as vectors (and as explored earlier in the vectors section) $$z - i$$ is the vector from $$i$$ to $$z$$ (which means that the length of the vector $$z - i$$ is the same as the distance between $$i$$ and $$z$$) so the absolute value of this is the distance between $$i$$ and $$z$$.