Mathematics Foundations/5.1 Definitions and Basic Properties

Complex Numbers 5.1 Definitions and Basic Properties 5.1.1 Introduction to Complex Numbers Complex numbers extend the real number system to include solutions to equations like , which have no solutions in the real number system. The complex number system provides a complete algebraic closure, meaning that every non-constant polynomial equation has a solution within this system. 5.1.2 The Imaginary Unit We define the imaginary unit as a number that satisfies: This definition allows us to express the solution to as . 5.1.3 Form of Complex Numbers A complex number is expressed in the form: where and are real numbers. We call: the real part of , denoted by the imaginary part of , denoted by Special cases: If , then is a real number If and , then is called a pure imaginary number 5.1.4 Equality of Complex Numbers Two complex numbers and are equal if and only if: This means both their real and imaginary parts must be equal. 5.1.5 Basic Arithmetic Operations For complex numbers and : Addition: Subtraction: Multiplication: This multiplication formula follows from the distributive property and the fact that : Division: Therefore: Note that division by is defined only when , meaning and are not both zero. 5.1.6 Complex Conjugate The complex conjugate of , denoted by or , is defined as: Properties of complex conjugates: (always a non-negative real number) 5.1.7 Modulus of a Complex Number The modulus or absolute value of a complex number , denoted by , is defined as: This represents the distance from the origin to the point in the complex plane. Properties of the modulus: for all , with if and only if for (Triangle inequality) 5.1.8 The Field of Complex Numbers The set of complex numbers, denoted by , forms a field under the operations of addition and multiplication. This means: is an abelian group with identity element 0 is an abelian group with identity element 1 Multiplication distributes over addition: The field of complex numbers contains the field of real numbers as a subfield. 5.1.9 Algebraic Completeness One of the most important properties of the complex number system is its algebraic completeness, formalized in the Fundamental Theorem of Algebra: Theorem 5.1.1 (Fundamental Theorem of Algebra): Every non-constant polynomial with complex coefficients has at least one complex root. This means that any polynomial equation of degree has exactly roots in the complex number system, counting multiplicities. 5.1.10 Powers of the Imaginary Unit The powers of follow a cyclic pattern: (the pattern repeats) In general, for any integer : 5.1.11 Historical Note Complex numbers were first introduced in the 16th century by Italian mathematicians, including Gerolamo Cardano, who used them to solve cubic equations. Initially met with skepticism and called "imaginary" or "impossible" numbers, they gradually gained acceptance as their mathematical utility became clear. The modern notation for the imaginary unit was introduced by Leonhard Euler in the 18th century. 5.1.12 Exercises 1. Compute the following: a) b) c) 2. Prove that for any complex numbers and . 3. Show that and . 4. Find all complex numbers such that . 5. Prove that .