1.3 The Geometry of Complex Numbers

    Complex numbers are ordered pairs of real numbers, so they can be represented by points in the plane. In this section we show the effect that algebraic operations on complex numbers have on their geometric representations.

    We can represent the number    by a position vector in the xy plane whose tail is at the origin and whose head is at the point  .  When the xy plane is used for displaying complex numbers, it is called the complex plane, or more simply, the plane.  Recall that  .  Geometrically,    is the projection of    onto the axis, and    is the projection of onto the axis. It makes sense, then, to call the axis the real axis and the axis the imaginary axis, as Figure 1.3 illustrates.

            Figure 1.3  The complex plane.

    Addition of complex numbers is analogous to addition of vectors in the plane.  As we saw in Section 1.2, the sum of and is .  Hence, can be obtained vectorially by using the "parallelogram law," where the vector sum is the vector represented by the diagonal of the parallelogram formed by the two original vectors. Figure 1.4 illustrates this notion.

            Figure 1.4  The sum .  

    The difference    can be represented by the displacement vector from the point    to the point  ,  as Figure 1.5 shows.

            Figure 1.5  The difference .  

 

Definition 1.8, (Modulus or Absolute Value).  The modulus, or absolute value, of the complex number    is a nonnegative real number denoted by    and is given by the equation  

(1-20)            .  

    The number    is the distance between the origin and the point  .  The only complex number with modulus zero is the number .  The number    has modulus  ,  and is depicted in Figure 1.6.  

            Figure 1.6  The real and imaginary parts of a complex number.

    The numbers    are the lengths of the sides of the right triangle OPQ shown in Figure 1.7.  

            Figure 1.7  The moduli of z and its components.

    The inequality means that the point is closer to the origin than the point .  Although obvious from Figure 1.7, it is still profitable to work out algebraically the standard results that    

(1-21)              and  .
            
which we leave as an exercise for the reader.

    The difference represents the displacement vector from to , so the distance between and is given by.  We can obtain this distance by using Definition (1.2) and Definition (1.8) to obtain the familiar formula

            .  

    If  ,  then    is the reflection of , through the origin, and    is the reflection of through the axis, as illustrated in Figure 1.8.

            Figure 1.8 The geometry of negation and conjugation.

 

    We can use an important algebraic relationship to establish properties of the absolute value that have geometric applications.  Its proof is rather straightforward, and we ask you to give it in the exercises for this section.
    
(1-22)             .  

    A beautiful and important application of the above identity is its use in establishing the triangle inequality, which states that the sum of the lengths of two sides of a triangle is greater than or equal to the length of the third side. Figure 1.9 illustrates this inequality.

            Figure 1.9  The triangle inequality.

Theorem 1.2, (Triangle Inequality).   If   are arbitrary complex numbers, then  

(1-23)             .  

Proof.  We appeal to basic results:  

Taking square roots yields the desired inequality.  

Example  1.5.  To produce an example of which Figure 1.9 is a reasonable illustration, we let  .  Then    and  .  Clearly,  ;  hence  .  In this case, we can verify the triangle inequality without recourse to computation of square roots because  

        ,  
thus
            .  

Explore Solution 1.5.

    We can also establish other important identities by means of the triangle inequality. Note that  
    
                  
                  
                .  

Subtracting from the left and right sides of this string of inequalities gives an important relationship that will be used in determining lower bounds of sums of complex numbers:

(1-24)             .  

Using the identity    and the commutative and associative laws it follows that  

(1-20)               
(1-20)               
(1-20)               

Taking square roots of the terms on the left and right establishes another important identity  

(1-25)             .  

As an exercise, we ask you to show

(1-26)             , provided   .  

Example  1.6.  Use the values  ,  then    and  .  Also  ;  hence  

            ,  
thus
            .  

Explore Solution 1.6.

    Figure 1.10 illustrates the multiplication shown in Example1.6.  The length of the vector apparently equals the product of the lengths of , confirming that  ,  but why is it located in the second quadrant when both are in the first quadrant?  The answer to this question will become apparent to you in Section 1.4.

            Figure 1.10  The geometry of multiplication.

Exercises for Section 1.3.  The Geometry of Complex Numbers  

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell