## 18.1 Notations, Representation of a Complex Number by Magnitude and Angle,
Real and Imaginary Parts

**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.