## Complex numbers

In this chapter we introduce an extension of real numbers called complex numbers. At first, it is difficult to understand the real-life importance of such an apparently abstract concept as complex numbers. However, there are deep mathematical reasons why complex numbers inevitably appear in many applications. For example, matrices with real entries can have complex eigenvalues. The behavior of such well-known functions as sine and cosine is properly understood only when we introduce complex numbers. Most importantly, one of the key questions in time-series analysis deals with the stability of the system. This question is reduced to finding eigenvalues of a certain matrix, which describe the ongoing autoregressive process.