Introduction to the complex-number system - Continuous and Discrete Time Signals and Systems - page 797

Appendix B Introduction to the

complex-number system

In this appendix, we introduce some elementary concepts that define complex
numbers. In presenting the material, it is anticipated that most readers have
some prior exposure to complex numbers, so the information presented here
serves primarily as a review. The appendix is organized as follows. In Section
B.1, we review the definition of real numbers and then survey their arithmetic
properties, including some basic operations like addition, subtraction, multipli-
cation, and division. Section B.2 extends the arithmetic operations to complex
numbers, and Section B.3 introduces its geometric structure using the 2D Carte-
sian representation. Section B.4 presents an alternative representation, referred
to as the polar representation for complex numbers. Section B.5 concludes the
appendix.

B.1 Real-number s ystem

A real-number system ℜ is a set of all real numbers, which is defined in terms of
two basic operations: addition and multiplication. For two arbitrarily selected
real numbers a , b ∈ℜ , these basic operations are given by

addition s 1 = a + b ; (B.1)

multiplication m 1 = a b , (B.2)

such that s 1 , m 1 ∈ℜ . The remaining arithmetic operations, for example, sub-
traction and division, are expressed in terms of Eqs (B.1) and (B.2) as follows:

subtraction s 2 = a − b = a + ( − b ); (B.3)

division m 2 = a / b = a (1 / b ) , (B.4)

such that s 2 , m 2 ∈ℜ . The real number − b is referred to as the additive inverse
of b since b + ( − b ) = 0. Likewise, the real number 1 / b is referred to as the
multiplicative inverse of b since b (1 / b ) = 1. For ℜ to represent a complete
set of real numbers, it must satisfy the following properties.

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