Introduction to Complex Numbers

10.6  Introduction to Complex Numbers

  When the index n in root(n,a); is even, a is restricted to the positive real numbers. In the real number system root(-2) is not defined. For the square root of a negative number to have meaning, a new unit, called the imaginary unit, root(-1) is introduced and is denoted by i. Since (root(a))^2 was defined to be a, for conformity i is defined so that i^2=-1.

DEFINITION  If a∈R,a>0, we define root(-a)=root(-1)root(a)=i√a.

  Any number of the form ai,a∈i=root(-1), is called a pure imaginary number.

EXAMPLES  1. root(-4)=root(-1)root(4)=i√4=2i

        2. root(-7)=root(-1)root(7)=i√7

        3. root(-12)=root(-1)root(12)=i(2root(3))=2i√3

  When a,b, and c are real numbers, a*b+c is also a real number. However, the expression ai*bi+ci=abi^2+ci=-ab+ci is neithera real number nor is it a pure imaginary number.

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DEFINITION  A complex number is a number of the form a+bi, where aand b are real numbers and i=root(-1). The number a is called the real part cf the complex number, and b is called the imaginary part.

  The set of complex numbers, denoted by C. is the set

  When a complex number is written in the form a+bi, the complex number is said to be in simplified form, or m standard form. The form a+bi is sometimes referred to as the Cartesian or rectangular form of a complex number.

Notes  1. The complex number a+0i=a is a real number. That is, the set of real numbers R is a subset of the set of complex numbers C.
     2. The complex number 0+bi,b!=0, is a pure imaginary number. That is, the set of pure imaginary numbers is a subset of the set of complex numbers.