1.2  The Algebra of Complex Numbers

    We have seen (in Section 1.1) that complex numbers came to be viewed as ordered pairs of real numbers. That is, a complex number is defined to be

            ,
        
where x and y are both real numbers.

    The reason we say ordered pair is because we are thinking of a point in the plane.  The point (2, 3), for example, is not the same as (3, 2). The order in which we write and in the equation makes a difference.  Clearly, then, two complex numbers are equal if and only if their x coordinates are and their y coordinates are equal. In other words,

               iff   .
        
(Throughout this text, iff means if and only if.)

    A meaningful number system requires a method for combining ordered pairs.  The definition of algebraic operations must be consistent so that the sum, difference, product, and quotient of any two ordered pairs will again be an ordered pair.  The key to defining how these numbers should be manipulated is to follow Gauss's lead and equate with .  Then, if and are arbitrary complex numbers, we have

    Thus, the following definitions should make sense.

Definition 1.1, (Addition)

Formula (1-8)    .  

Derivation of  Formula (1-8).

Definition 1.2, (Subtraction)

Formula (1-9)    .  

Derivation of  Formula (1-9).

Example 1.1.  Given  .  (a)  Find    and  (b)  .  

              and  
        
            .

We can also use the notation and :

              and  

            .

Explore Solution 1.1. (a).

Explore Solution 1.1. (b).

    Given the rationale we devised for addition and subtraction, it is tempting to define the product    as  .  It turns out, however, that this is not a good definition, and we ask you in the exercises for this section to explain why.  How, then, should products be defined?  Again, if we equate    with    and assume, for the moment, that    makes sense (so that  ), we have

    Thus, it appears we are forced into the following definition.

Definition 1.3, (Complex Multiplication)

Formula (1-10)    .  

Derivation of  Formula (1-10).

Example 1.2.  Given  .  Find  .  
    
            

We get the same answer by using the notation and :

            

Of course, it makes sense that the answer came out as we expected because we used the notation x+iy as motivation for our definition in the first place.

Explore Solution 1.2.

    To motivate our definition for division, we will proceed along the same lines as we did for multiplication, assuming :  

              

    
We need to figure out a way to write the preceding quantity in the form  .  To do this, we use a standard trick and multiply the numerator and denominator by  ,  which gives

               

Thus, we finally arrive at a rather odd definition.

Definition 1.4, (Complex Division)

Formula (1-11)    ,   for   .  

Derivation of  Formula (1-11).

Example 1.3.  Given  .  Find  .  

            
    
As with the example for multiplication, we also get this answer if we use the notation  :

              

Explore Solution 1.3.

    To perform operations on complex numbers, most mathematicians would use the notation    and engage in algebraic manipulations, as we did here, rather than apply the complicated-looking definitions we gave for those operations on ordered pairs.  This procedure is valid because we used the    notation as a guide for defining the operations in the first place.  Remember, though, that the    notation is nothing more than a convenient bookkeeping device for keeping track of how to manipulate ordered pairs.  It is the ordered pair algebraic definitions that form the real foundation on which the complex number system is based.  In fact, if you were to program a computer to do arithmetic on complex numbers, your program would perform calculations on ordered pairs, using exactly the definitions that we gave.

    It turns out that our algebraic definitions give complex numbers all the properties we normally ascribe to the real number system.  Taken together, they describe what algebraists call a field.  In formal terms, a field is a set (in this case, the complex numbers) together with two binary operations (in this case, addition and multiplication) having the following properties.

(P1) Commutative Law for Addition.  

            .  

Derivation of  Property (P1).

(P2) Associative Law for Addition.  

            .  

(P3) Additive Identity.  There is a complex number such that

             for all complex numbers z.
        
The number is obviously the ordered pair .

(P4) Additive Inverses.  Given any complex number , there is a complex number (depending on ) with the property that

            .

Obviously, if , the number will be .

(P5) Commutative Law for Multiplication.   

            .  

(P6) Associative Law for Multiplication.  

            .  

(P7) Multiplicative Identity.  There is a complex number such that

             for all complex numbers .

As one might expect, it turns out that is the unique complex number with this property. We ask you to verify this identity in the exercises for this section.

(P8) Multiplicative Inverses.  Given any number other than the number (0, 0), there is a complex number (depending on z) which we shall denote by with the property that

            .

Based on our definition for division, it seems reasonable that the number would be  

            .  

We ask you to confirm this result in the exercises for this section.

(P9) The Distributive Law.  

            .  

Derivation of  Property (P 9).

    None of these properties is difficult to prove.  Most of the proofs make use of corresponding facts in the real number system. To illustrate, we give a proof of property (P1).

Proof of the commutative law for addition:  Let and be arbitrary complex numbers. Then,

            

    Actually, you can think of the real number system as a subset of the complex number system.  To see why, let's agree that, as any complex number of the form is on the axis, we can identify it with the real number .  With this correspondence, we can easily verify that our definitions for addition, subtraction, multiplication, and division of complex numbers are consistent with the corresponding operations on real numbers.  For example, if and are real numbers, then

            

    It is now time to show specifically how the symbol relates to the quantity  .  Note that

            

    If we use the symbol for the point , the preceding identity gives

            ,

which means  .  So, the next time you are having a discussion with your friends and they scoff when you claim that is not imaginary, calmly put your pencil on the point of the coordinate plane and ask them if there is anything imaginary about it.  When they agree there isn't, you can tell them that this point, in fact, represents the mysterious in the same way that represents .

    We can also see more clearly now how the notation    quates to .  Using the preceding conventions (i.e.,  , etc.), we have  

              

    Thus we may move freely between the notations    and , depending on which is more convenient for the context in which we are working.  Students sometimes wonder whether it matters where the " is located in writing a complex number.  It does not.  Generally, most texts place terms containing an at the end of an expression, and place the " before a variable, but after a constant.  Thus, we write  ,  , etc.,  but  ,    and so forth.  Because letters lower in the alphabet generally denote constants, you will usually (but not always) see the expression  ,  instead of  .  Many authors write quantities like   instead of    to make sure the " is not mistakenly thought to be inside the square root symbol.  Additionally, if there is concern that the " might be missed, it is sometimes placed before a lengthy expression, as in  .

    We close this section with three important definitions and a theorem involving them.  We ask you for a proof of the theorem in the excercises.
 

Definition 1.5, (Real Part of z).  The real part of  ,  denoted by  ,  is the real number  .

Definition 1.6, (Imaginary Part of z).  The imaginary part of  ,  denoted by  ,  is the real number  .

Definition 1.7, (Conjugate of z).  The conjugate of  ,  denoted by  ,  is the complex number  .

Example 1.4.  Given  .   

1.4 (a)  We have       and    .  

1.4 (b)  We have       and    .  

1.4 (c)  We have       and      .  

Explore Solution 1.4 (a).

Explore Solution 1.4 (b).

Explore Solution 1.4 (c).

    The following theorem gives some important facts relating to these operations. You will be asked for a proof in the exercises

Theorem 1.1.  Suppose are arbitrary complex numbers. Then if

    Because of what it erroneously connotes, it is a shame that the term imaginary is used in Definition (1.6).  It was coined by the brilliant mathematician and philosopher René Descartes (1596--1650) during an era when quantities such as were thought to be just that.  Gauss, who was successful in getting mathematicians to adopt the phrase complex number rather than imaginary number, also suggested that they use lateral part of z in place of imaginary part of z.  Unfortunately, that suggestion never caught on, and it appears we are stuck with what history has handed down to us.

Exercises for Section 1.2.  The Algebra of Complex Numbers