Why study complex numbers, and functions of of a complex variable? Answer
We introduce i, the square root of -1, so as to allow negative numbers to have square roots, something they do not have among ordinary real numbers.
Complex numbers can be described as vectors in two dimensional Euclidean space.
We normally use the x variable to represent the real part of the number and the y variable to represent its imaginary part. Thus the basis vectors i and j when dealing with complex numbers are the numbers 1 and i, respectively.
The number (1 + i) can then be represented as the vector (1, 1).
In this context the length of the vector, r is the positive square root of the sum of the components.
For a number (a + ib), r is the square root of (a + ib)(a - ib).
The angle is defined exactly as for vectors.
We usually refer to the x component of this vector as its real part, and the y component as its imaginary part.
Complex numbers have the additional property, that ordinary vectors lack, that we can define multiplication among them so as to obey the usual commutative, associative and distributive laws of arithmetic. This fact allows definitions of complex valued functions by the same sort of rules that we use to define ordinary real functions.