Chapter 1  Complex Numbers

Prelude

    A practical use of complex analysis is finding the roots of a polynomial. Our modules start with the origins of complex analysis and develop the theory that is necessary to understand present day software and applications. Along the way we will be using modern software tools.  For example,

                

The graph on the left is used in calculus.  The others involve the complex argument, Arg[p (z)];  and the absolute value, |p (z)|.   
Before we get around to these concepts we will be reviewing the origins of complex analysis and other essential foundations.

Overview of Chapter 1

    Get ready for a treat. You're about to begin studying some of the most beautiful ideas in mathematics. They are ideas with surprises. They evolved over several centuries, yet they greatly simplify extremely difficult computations, making some as easy as sliding a hot knife through butter. They also have applications in a variety of areas, ranging from fluid flow, to electric circuits, to the mysterious quantum world. Generally, they are described as belonging to the area of mathematics known as complex analysis.

Section 1.1  The Origin of Complex Numbers

    Complex analysis can roughly be thought of as the subject that applies the theory of calculus to imaginary numbers. But what exactly are imaginary numbers? Usually, students learn about them in high school with introductory remarks from their teachers along the following lines: "We can't take the square root of a negative number. But let's pretend we can and begin by using the symbol ." Rules are then learned for doing arithmetic with these numbers. At some level the rules make sense. If , it stands to reason that . However, it is not uncommon for students to wonder whether they are really doing magic rather than mathematics.

    If you ever felt that way, congratulate yourself! You're in the company of some of the great mathematicians from the sixteenth through the nineteenth centuries. They, too, were perplexed by the notion of roots of negative numbers. Our purpose in this section is to highlight some of the episodes in the very colorful history of how thinking about imaginary numbers developed. We intend to show you that, contrary to popular belief, there is really nothing imaginary about "imaginary numbers." They are just as real as "real numbers."

    Our story begins in 1545. In that year the Italian mathematician Girolamo Cardano published Ars Magna (The Great Art), a 40-chapter masterpiece in which he gave for the first time an algebraic solution to the general cubic equation  

            .

    Cardano did not have at his disposal the power of today's algebraic notation, and he tended to think of cubes or squares as geometric objects rather than algebraic quantities.  Essentially, however, his solution began with the substiution .  This move transforms    into the cubic equation    without a squared term, which is called a depressed cubic and can be written as

            .

You need not worry about the computational details, but the coefficients are    and  .
Exploration.

    To illustrate, begin with    and substitute  .  The equation then becomes  , which simplifies to  .
Exploration.

    If Cardano could get any value of x that solved a depressed cubic, he could easily get a corresponding solution to from the identity . Happily, Cardano knew how to solve a depressed cubic. The technique had been communicated to him by Niccolo Fontana who, unfortunately, came to be known as Tartaglia (the stammerer) due to a speaking disorder. The procedure was also independently discovered some 30 years earlier by Scipione del Ferro of Bologna. Ferro and Tartaglia showed that one of the solutions to the depressed cubic equation is

        .  

    Although Cardano would not have reasoned in the following way, today we can take this value for x and use it to factor the depressed cubic into a linear and quadratic term. The remaining roots can then be found with the quadratic formula.

    For example, to solve  ,  use the substitution    to get  ,  which is a depressed cubic equation.  Next, apply the "Ferro-Tartaglia" formula with and to get  .  Since    is a root,     must be a factor of  .  Dividing     into    gives  ,  which yields the remaining (duplicate) roots of   .  The solutions to    are obtained by recalling  , which yields the three roots    and  .
Exploration.

    So, by using Tartaglia's work and a clever transformation technique, Cardano was able to crack what had seemed to be the impossible task of solving the general cubic equation.  Surprisingly, this development played a significant role in helping to establish the legitimacy of imaginary numbers.  Roots of negative numbers, of course, had come up earlier in the simplest of quadratic equations, such as  .  The solutions we know today as  , however, were easy for mathematicians to ignore.  In Cardano's time, negative numbers were still being treated with some suspicion, as it was difficult to conceive of any physical reality corresponding to them. Taking square roots of such quantities was surely all the more ludicrous. Nevertheless, Cardano made some genuine attempts to deal with . Unfortunately, his geometric thinking made it hard to make much headway. At one point he commented that the process of arithmetic that deals with quantities such as "involves mental tortures and is truly sophisticated." At another point he concluded that the process is "as refined as it is useless." Many mathematicians held this view, but finally there was a breakthrough.

    In his 1572 treatise L'Algebra, Rafael Bombelli showed that roots of negative numbers have great utility indeed. Consider the depressed cubic .  Using and in the "Ferro-Tartaglia" formula for the depressed cubic, we compute , or in a somewhat different form, .

    Simplifying this expression would have been very difficult if Bombelli had not come up with what he called a "wild thought."  He suspected that if the original depressed cubic had real solutions, then the two parts of x in the preceding equation could be written as    and    for some real numbers u and v.  That is, Bombeli believed   and  ,  which would mean    and  .  Then, using the well-known algebraic identity , and (letting and ),  and assuming that roots of negative numbers obey the rules of algebra, he obtained  
    
       
       
                                
                                
                              .  

    By equating like parts, Bombelli reasoned that    and  .  Perhaps thinking even more wildly, Bombelli then supposed that u and v were integers.  The only integer factors of 2 are 2 and 1, so the equation     led Bombelli to conclude that    and  .  From this conclusion it follows that  , or  . Amazingly, and   solve the second equation , so Bombelli declared the values for u and v to be u=2 and v=1, respectively.

    Since , we clearly have .  Similarly, Bombelli showed that . But this means that

    ,

which was a proverbial bombshell.  Prior to Bombelli, mathematicians could easily scoff at imaginary numbers when they arose as solutions to quadratic equations.  With cubic equations, they no longer had this luxury. That was a correct solution to the equation    was indisputable, as it could be checked easily.  However, to arrive at this very real solution, mathematicians had to take a detour through the uncharted territory of "imaginary numbers."  Thus, whatever else might have been said about these numbers (which, today, we call complex numbers), their utility could no longer be ignored.
Exploration.

    Admittedly, Bombelli's technique applies only to a few specialized cases, and lots of work remained to be done even if Bombelli's results could be extended.  After all, today we represent real numbers geometrically on the number line.  What possible representation could complex numbers have?  In 1673 John Wallis made a stab at a geometric picture of complex numbers that comes close to what we use today.  He was interested in representing solutions to general quadratic equations, which we can write as to make the following discussion easier to follow.  When we use the quadratic formula with this equation, we get

         and .

    Wallis imagined these solutions as displacements to the left and right from the point . He saw each displacement, whose value is , as the length of the sides of the right triangles shown in Figure 1.1. The points P � and P ‚ represent the solutions to our equation, which is clearly correct if  .  But how should we picture P � and P ‚ when negative roots arise (i.e., when )?  Wallis reasoned that, with negative roots, b would be less than c, so the lines of length b in Figure 1.1 would no longer be able to reach all the way to the x axis.  Instead, they would stop somewhere above it, as Figure 1.2 shows.  Wallis argued that P � and P ‚ should represent the geometric locations of the solutions    and     when  .  He evidently thought that, because b is shorter than c, it could no longer be the hypotenuse of the right triangle as it had been earlier. The side of length c would now have to take that role.

    Wallis's method has the undesirable consequence that    is represented by the same point as is  . Nevertheless, this interpretation helped set the stage for thinking of complex numbers as "points on the plane." By 1732, the great Swiss mathematician  Leonhard Euler  (pronounced "oiler") adopted this view concerning the n solutions to the equation  .  You will learn shortly that these solutions can be expressed as    for various values of  ;  Euler thought of them as being located at the vertices of a regular polygon in the plane. Euler was also the first to use the symbol for . Today, this notation is still the most popular, although some electrical engineers prefer the symbol instead so that they can use to represent current.

    Is it possible to modify slightly Wallis's picture of complex numbers so that it is consistent with the representation used today?  To help you answer this question, refer to the article by Alec Norton and Benjamin Lotto, "Complex Roots Made Visible," The College Mathematics Journal, 15(3), June 1984, pp. 248--249, Jstor.

    Two additional mathematicians deserve mention. The Frenchman Augustin-Louis Cauchy (1789--1857) formulated many of the classic theorems that are now part of the corpus of complex analysis.  The German Carl Friedrich Gauss (1777--1855) reinforced the utility of complex numbers by using them in his several proofs of the fundamental theorem of algebra (see Chapter 6).  In an 1831 paper, he produced a clear geometric representation of x+iy by identifying it with the point (x, y) in the coordinate plane. He also described how to perform arithmetic operations with these new numbers.

    It would be a mistake, however, to conclude that in 1831 complex numbers were transformed into legitimacy. In that same year the prolific logician Augustus De Morgan commented in his book, On the Study and Difficulties of Mathematics, "We have shown the symbol to be void of meaning, or rather self-contradictory and absurd. Nevertheless, by means of such symbols, a part of algebra is established which is of great utility."

    There are, indeed, genuine logical problems associated with complex numbers. For example, with real numbers    so long as both sides of the equation are defined. Applying this identity to complex numbers leads to 1=√1=√((-1)(-1))=√(-1)√(-1)=-1.  Plausible answers to these problems can be given, however, and you will learn how to resolve this apparent contradiction in Section 2.4. De Morgan's remark illustrates that many factors are needed to persuade mathematicians to adopt new theories. In this case, as always, a firm logical foundation was crucial, but so, too, was a willingness to modify some ideas concerning certain well-established properties of numbers.

    As time passed, mathematicians gradually refined their thinking, and by the end of the nineteenth century complex numbers were firmly entrenched. Thus, as it is with many new mathematical or scientific innovations, the theory of complex numbers evolved by way of a very intricate process. But what is the theory that Tartaglia, Ferro, Cardano, Bombelli, Wallis, Euler, Cauchy, Gauss, and so many others helped produce? That is, how do we now think of complex numbers? We explore this question in the remainder of this chapter.

The solution of the cubic equations.  Mathematica can construct the solutions to the general cubic equation. However they are quite formidable.
Exploration.

Exercises for Section 1.1.  The Origin of Complex Numbers