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 \(z\).