Module

for

The Origin of Complex Numbers

 

Complex Analysis for Mathematics and Engineering
by  John H. Mathews  and  Russell W. Howell

 

Chapter 1  Complex Numbers

Preliminary Remarks

        Complex numbers are introduced in algebra courses and the letter  [Graphics:Images/ComplexNumberOrigin_gr_1.gif]  is used to denote  [Graphics:Images/ComplexNumberOrigin_gr_2.gif],  and it is called

the imaginary unit. Since [Graphics:Images/ComplexNumberOrigin_gr_3.gif], it is the solution of the equation [Graphics:Images/ComplexNumberOrigin_gr_4.gif]. Then the notation [Graphics:Images/ComplexNumberOrigin_gr_5.gif] is introduced

and the sum, product and quotient of complex numbers are taught.

        The quadratic formula is another fact from algebra.  Given the quadratic equation  

                    [Graphics:Images/ComplexNumberOrigin_gr_6.gif],  

the roots are

                    [Graphics:Images/ComplexNumberOrigin_gr_7.gif].

        Then cubic equations are introduced.  Methods for solving a cubic equation require one real root [Graphics:Images/ComplexNumberOrigin_gr_8.gif] to be given

or easy to find.  For example, when students are asked to solve the cubic equation  

                    [Graphics:Images/ComplexNumberOrigin_gr_9.gif],  

the hint that the real root is  [Graphics:Images/ComplexNumberOrigin_gr_10.gif] might be given, or it might be left the student to discover.   Then next step is

to factor the cubic as

                    [Graphics:Images/ComplexNumberOrigin_gr_11.gif].   

To solve the equation  [Graphics:Images/ComplexNumberOrigin_gr_12.gif],  the method of factoring can be used or the quadratic formula can be used.  

The solutions to

                    [Graphics:Images/ComplexNumberOrigin_gr_13.gif]   

are easily found to be   [Graphics:Images/ComplexNumberOrigin_gr_14.gif].  

Then the three solutions to  [Graphics:Images/ComplexNumberOrigin_gr_15.gif]  are given by

                    [Graphics:Images/ComplexNumberOrigin_gr_16.gif].

Aside.  We can use Mathematica or Maple to plot the roots of   [Graphics:Images/ComplexNumberOrigin_gr_17.gif].  This is just for fun!

                                        [Graphics:Images/ComplexNumberOrigin_gr_18.gif]

  

Aside.  We can draw some more graphs of the polynomial.   This is just for fun!

 

[Graphics:Images/ComplexNumberOrigin_gr_19.gif][Graphics:Images/ComplexNumberOrigin_gr_20.gif]

                                   [Graphics:Images/ComplexNumberOrigin_gr_21.gif]                                                                                          [Graphics:Images/ComplexNumberOrigin_gr_22.gif]

 

[Graphics:Images/ComplexNumberOrigin_gr_23.gif][Graphics:Images/ComplexNumberOrigin_gr_24.gif]

                                   [Graphics:Images/ComplexNumberOrigin_gr_25.gif]                                                                             [Graphics:Images/ComplexNumberOrigin_gr_26.gif]

                                                                      Graphs of the cubic polynomial  [Graphics:Images/ComplexNumberOrigin_gr_27.gif].  

                                                            Can you locate the roots  [Graphics:Images/ComplexNumberOrigin_gr_28.gif] ?

Explore the Details.

 

Remarks.   We will introduce the notation  [Graphics:Images/ComplexNumberOrigin_gr_97.gif]  in Section 1.2.   

We will discuss  [Graphics:Images/ComplexNumberOrigin_gr_98.gif],  the absolute value (or modulus)  in Section 1.3,

and the argument  [Graphics:Images/ComplexNumberOrigin_gr_99.gif] in Section 1.4.  

We will introduce,   [Graphics:Images/ComplexNumberOrigin_gr_100.gif],   a function of the complex variable  [Graphics:Images/ComplexNumberOrigin_gr_101.gif],  in Section 2.1.  

        The solution to the general cubic equation   [Graphics:Images/ComplexNumberOrigin_gr_102.gif]   is introduced in calculus where we

usually say that the formula is very complicated and that Newton's method for finding a numerical approximation

should be used.  Then the computer algorithm for Newton's method is taught, and students are asked to use

computer software or programmable calculator.

        It may come as a surprise that centuries ago, the attempt to solve cubic equations led to the invention

of complex numbers.

 

Overview

        Get ready for a treat. You're about to begin studying some of the most beautiful ideas in mathematics. They

are ideas with surprises, and 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.

        If you prefer, you can chose to read this section at a later time.  

The main part of this course starts with Section 1.2.  

 

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.  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  [Graphics:Images/ComplexNumberOrigin_gr_103.gif]."  Rules are then learned for doing arithmetic with

these numbers.  At some level the rules make sense.  

If  [Graphics:Images/ComplexNumberOrigin_gr_104.gif],  it stands to reason that  [Graphics:Images/ComplexNumberOrigin_gr_105.gif].  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."

     To prepare our thinking, let us focus on the following special way to solve   [Graphics:Images/ComplexNumberOrigin_gr_106.gif].  

Since the equation is a quadratic, we must have  [Graphics:Images/ComplexNumberOrigin_gr_107.gif],  and we can divide each term by  [Graphics:Images/ComplexNumberOrigin_gr_108.gif]  and get,

                    [Graphics:Images/ComplexNumberOrigin_gr_109.gif].  
                    
Now use the special substitution  [Graphics:Images/ComplexNumberOrigin_gr_110.gif]  and get,  

                    [Graphics:Images/ComplexNumberOrigin_gr_111.gif].    

which can be simplified to obtain the depressed quadratic equation,  

                    [Graphics:Images/ComplexNumberOrigin_gr_112.gif].  

It is an easy task solve this depressed quadratic equation, because there is no linear term  [Graphics:Images/ComplexNumberOrigin_gr_113.gif],  

and all we need to do is move the constant term to the right side of the equation, and solve it,

                    [Graphics:Images/ComplexNumberOrigin_gr_114.gif].  

then take the square root and obtain the two solutions,  

                    [Graphics:Images/ComplexNumberOrigin_gr_115.gif],  

and  
                    [Graphics:Images/ComplexNumberOrigin_gr_116.gif].  

     Then recall the special substitution that we used  [Graphics:Images/ComplexNumberOrigin_gr_117.gif],  

construct the two solutions to the given quadratic equation   [Graphics:Images/ComplexNumberOrigin_gr_118.gif],  

                    [Graphics:Images/ComplexNumberOrigin_gr_119.gif],  
and  
                    [Graphics:Images/ComplexNumberOrigin_gr_120.gif],  

which are the solutions we want.

     Four sixteenth century Italian mathematicians made significant contributions leading up to the invention of complex

numbers:  Scipione del Ferro (1465-1526),  Nicolo Tartaglia (1500-1557),  Girolamo Cardano (1501-1576),

and  Rafael Bombelli (1526-1572).  

        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  

                    [Graphics:Images/ComplexNumberOrigin_gr_121.gif].

        Cardano did not have at his disposal the power of today's algebraic notation, (and computers)  and his

computations were limited to numbers in "real domain" (also a Maple computing environment).  Cardano tended

to think of cubes or squares as geometric objects rather than algebraic quantities.  However, he is credited for

making the following important discovery.  

 

Cardano's Substitution.  Given a general cubic equation

                    [Graphics:Images/ComplexNumberOrigin_gr_122.gif].  

If you make the substitution  [Graphics:Images/ComplexNumberOrigin_gr_123.gif],  this will transform the general cubic equation into  

                    [Graphics:Images/ComplexNumberOrigin_gr_124.gif],    

without a squared term. This form is called the depressed cubic equation.  

You need not worry about the details, but the coefficients are  [Graphics:Images/ComplexNumberOrigin_gr_125.gif]  and  [Graphics:Images/ComplexNumberOrigin_gr_126.gif].

Exploration for Cardano's Substitution.

 

Cardano's Example 1.  Given the cubic equation

                    [Graphics:Images/ComplexNumberOrigin_gr_183.gif].  

If you make the substitution  [Graphics:Images/ComplexNumberOrigin_gr_184.gif],  this will transform the cubic equation into  

                    [Graphics:Images/ComplexNumberOrigin_gr_185.gif],  

which simplifies to  

                    [Graphics:Images/ComplexNumberOrigin_gr_186.gif].  

Exploration for Cardano's Example 1.

 

The Ferro-Tartaglia-Cardano Formula  

     If Cardano could get any value [Graphics:Images/ComplexNumberOrigin_gr_215.gif] that solved a depressed cubic, he could easily construct a solution to  

                    [Graphics:Images/ComplexNumberOrigin_gr_216.gif],

by using the substitution  [Graphics:Images/ComplexNumberOrigin_gr_217.gif].  Happily, Cardano knew how to solve a depressed cubic.  The technique

had been communicated to him by Niccolo Fontana who came to be known as Tartaglia (the stammerer)

due to a speaking disorder.  This procedure was also independently discovered some 30 years earlier by

Scipione del Ferro of Bologna.  Ferro, Tartaglia and Cardano found one solution  [Graphics:Images/ComplexNumberOrigin_gr_218.gif]  to the depressed cubic.

Theorem (Ferro-Tartaglia-Cardano Cubic Formula).  One solution to the depressed cubic equation  

                    [Graphics:Images/ComplexNumberOrigin_gr_219.gif],  
is
                    [Graphics:Images/ComplexNumberOrigin_gr_220.gif].  

Explore the Ferro-Tartaglia-Cardano Formula.

 

     Several books that say complex numbers first came up in the context of solving quadratic equations, but  this

is not true.  However it was Bombelli, who was the pioneer when he considered the case [Graphics:Images/ComplexNumberOrigin_gr_286.gif] in the

Ferro-Tartaglia-Cardano formula, and was forced to consider the possibility that there are imaginary numbers.  

(see, "Mathematics and Its History", by  John Stillwell,  Springer,  New York,  2010).  

        Although Cardano would not have reasoned in the following way, today we can take this value for  [Graphics:Images/ComplexNumberOrigin_gr_287.gif]  and

use it to factor the depressed cubic into a linear and quadratic term.  The remaining two roots,  [Graphics:Images/ComplexNumberOrigin_gr_288.gif],  

can then be found with the quadratic formula.

Cardano's Example 2.   Solve the cubic equation  

                    [Graphics:Images/ComplexNumberOrigin_gr_289.gif].

Cardano's Solution 2.   

To solve  [Graphics:Images/ComplexNumberOrigin_gr_290.gif],  notice that [Graphics:Images/ComplexNumberOrigin_gr_291.gif] and use the substitution  [Graphics:Images/ComplexNumberOrigin_gr_292.gif]  and get

                    [Graphics:Images/ComplexNumberOrigin_gr_293.gif]

Hence,
                    [Graphics:Images/ComplexNumberOrigin_gr_294.gif],  

is the corresponding depressed cubic equation.  

Next, apply the "Ferro-Tartaglia" formula with   [Graphics:Images/ComplexNumberOrigin_gr_295.gif]   and   [Graphics:Images/ComplexNumberOrigin_gr_296.gif]   and calculate  

                    [Graphics:Images/ComplexNumberOrigin_gr_297.gif]  

Hence,  

                    [Graphics:Images/ComplexNumberOrigin_gr_298.gif]  

is a root.  

Next, divide   [Graphics:Images/ComplexNumberOrigin_gr_299.gif]   into   [Graphics:Images/ComplexNumberOrigin_gr_300.gif]   and get  

                    

[Graphics:Images/ComplexNumberOrigin_gr_301.gif]

  

Thus,  

                    [Graphics:Images/ComplexNumberOrigin_gr_302.gif].  

Hence,  [Graphics:Images/ComplexNumberOrigin_gr_303.gif]  is a factor of the depressed cubic  [Graphics:Images/ComplexNumberOrigin_gr_304.gif]  and we have   

                    [Graphics:Images/ComplexNumberOrigin_gr_305.gif].  

Now it is easy to see that the remaining (duplicate) roots of  [Graphics:Images/ComplexNumberOrigin_gr_306.gif]  are  [Graphics:Images/ComplexNumberOrigin_gr_307.gif].

Therefore, the solutions to  

                    [Graphics:Images/ComplexNumberOrigin_gr_308.gif],  

are obtained by recalling the substitution  [Graphics:Images/ComplexNumberOrigin_gr_309.gif],  which yields  

                    [Graphics:Images/ComplexNumberOrigin_gr_310.gif]

Exploration for Cardano's Example 2.

 

Example 3.   Given depressed cubic equation   

                    [Graphics:Images/ComplexNumberOrigin_gr_404.gif].  

Use the Ferro-Tartaglia method to find one real root.  

Explore Example 3.

 

Example 4.   Given depressed cubic equation   

                    [Graphics:Images/ComplexNumberOrigin_gr_488.gif].  

Use the Ferro-Tartaglia method to find one real root.  

Explore Example 4.

 

        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  [Graphics:Images/ComplexNumberOrigin_gr_576.gif].  The solutions we know today as

[Graphics:Images/ComplexNumberOrigin_gr_577.gif] 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 [Graphics:Images/ComplexNumberOrigin_gr_578.gif]. 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  [Graphics:Images/ComplexNumberOrigin_gr_579.gif] "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.  

 

Bombelli's Example 5.   Solve the depressed cubic equation  

                    [Graphics:Images/ComplexNumberOrigin_gr_580.gif].  

Bombelli's Solution 5.   

Bombelli then used the "Ferro-Tartaglia" formula for finding a solution to   [Graphics:Images/ComplexNumberOrigin_gr_581.gif],  which is  


                    [Graphics:Images/ComplexNumberOrigin_gr_582.gif].  

To solve   [Graphics:Images/ComplexNumberOrigin_gr_583.gif],   he substituted   [Graphics:Images/ComplexNumberOrigin_gr_584.gif]  and  [Graphics:Images/ComplexNumberOrigin_gr_585.gif]   and computed   


                    [Graphics:Images/ComplexNumberOrigin_gr_586.gif]   
                    

        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  [Graphics:Images/ComplexNumberOrigin_gr_587.gif]

in the preceding equation could be written as  

                    [Graphics:Images/ComplexNumberOrigin_gr_588.gif]    and    [Graphics:Images/ComplexNumberOrigin_gr_589.gif]  

for some real numbers  [Graphics:Images/ComplexNumberOrigin_gr_590.gif].   That is, Bombelli believed   

                     [Graphics:Images/ComplexNumberOrigin_gr_591.gif]    and    [Graphics:Images/ComplexNumberOrigin_gr_592.gif],  

which would mean  

                    [Graphics:Images/ComplexNumberOrigin_gr_593.gif]    and   [Graphics:Images/ComplexNumberOrigin_gr_594.gif].  

Then, using the well-known algebraic identity  [Graphics:Images/ComplexNumberOrigin_gr_595.gif],   (letting  [Graphics:Images/ComplexNumberOrigin_gr_596.gif]  and  [Graphics:Images/ComplexNumberOrigin_gr_597.gif]),

and assuming that roots of negative numbers obey the rules of algebra, he expanded  [Graphics:Images/ComplexNumberOrigin_gr_598.gif]  and obtained

                    [Graphics:Images/ComplexNumberOrigin_gr_599.gif]  

        By equating like parts, in the last two lines,  [Graphics:Images/ComplexNumberOrigin_gr_600.gif],   

Bombelli reasoned that  

                    [Graphics:Images/ComplexNumberOrigin_gr_601.gif],  
and  
                    [Graphics:Images/ComplexNumberOrigin_gr_602.gif].   

Perhaps thinking even more wildly, Bombelli then supposed that [Graphics:Images/ComplexNumberOrigin_gr_603.gif] should be integers.  The only integer

factors of  [Graphics:Images/ComplexNumberOrigin_gr_604.gif],  so in the first equation,   [Graphics:Images/ComplexNumberOrigin_gr_605.gif],   Bombelli concluded that  

                    [Graphics:Images/ComplexNumberOrigin_gr_606.gif]    and    [Graphics:Images/ComplexNumberOrigin_gr_607.gif].  

Using his conclusion, [Graphics:Images/ComplexNumberOrigin_gr_608.gif] becomes  [Graphics:Images/ComplexNumberOrigin_gr_609.gif] which simplifies as [Graphics:Images/ComplexNumberOrigin_gr_610.gif]. Then it follows that

                    [Graphics:Images/ComplexNumberOrigin_gr_611.gif]    and    [Graphics:Images/ComplexNumberOrigin_gr_612.gif],  

and there are two choices [Graphics:Images/ComplexNumberOrigin_gr_613.gif]. Amazingly, he also found that [Graphics:Images/ComplexNumberOrigin_gr_614.gif] are solutions to the second equation

                    [Graphics:Images/ComplexNumberOrigin_gr_615.gif],

so Bombelli declared that the values for  [Graphics:Images/ComplexNumberOrigin_gr_616.gif]  should be  [Graphics:Images/ComplexNumberOrigin_gr_617.gif],  respectively.

     Using these values in his equation   [Graphics:Images/ComplexNumberOrigin_gr_618.gif],   he could now declare that

                    [Graphics:Images/ComplexNumberOrigin_gr_619.gif],

and then he could take the cube root and write  

                    [Graphics:Images/ComplexNumberOrigin_gr_620.gif].

By a similar line of reasoning, it follows that  

                    [Graphics:Images/ComplexNumberOrigin_gr_621.gif].   

Then, this implies that  

                    [Graphics:Images/ComplexNumberOrigin_gr_622.gif],

which was a proverbial bombshell.  Moreover, Bombelli did it all without knowing the modern interpretation

of the symbol "[Graphics:Images/ComplexNumberOrigin_gr_623.gif]" that was yet to come.  Complex numbers did not exist in Bombelli's world!

Explore Bombelli's Solution 5.

 

Example 6.   Use Bombelli's method to find the integer solution  [Graphics:../Images/ComplexNumberOrigin_gr_843.gif]  to the depressed cubic equation,   

                    [Graphics:Images/ComplexNumberOrigin_gr_731.gif].  

Explore Example 6.

 

Example 7.   Use Bombelli's method to find the integer solution  [Graphics:../Images/ComplexNumberOrigin_gr_981.gif]  to the depressed cubic equation,   

                    [Graphics:Images/ComplexNumberOrigin_gr_893.gif].  

Explore Example 7.

 

        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 cavalier attitude.  That  [Graphics:Images/ComplexNumberOrigin_gr_1065.gif]  was a correct

solution to the equation  [Graphics:Images/ComplexNumberOrigin_gr_1066.gif]  was indisputable.  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.  This was the dawning of a new era in mathematics, the "Age of complex numbers."

 

John Wallis leads the way to embed the real numbers in the plane.

Geometric Progress of John Wallis

        As significant as Bombelli's work was his results left many issues unresolved.  For example, his technique

applied only to a few specialized cases.  Could it be extended?  Even if it could be extended a larger question

remained: what possible physical representation could complex numbers have?

        The last question remained unanswered for more than two centuries; University of New Hampshire professor

Paul J. Nahin describes the progress  as occurring in several stages (see "An Imaginary tale: the Story of [Graphics:Images/ComplexNumberOrigin_gr_1067.gif]",

by Paul J Nahin, Princeton University Press, 2007, pages 48-55).  

        
     The seventeenth century English mathematician John Wallis (1616-1703) also made a contribution.  

A preliminary step came in 1685 when the English mathematician John Wallis published "A treatise of Algebra,

both Historical and Practical
."
  Among the many contributions in that book two are particularly noteworthy

for our purposes.  They are displayed in Wallis' analysis of a problem from classical geometry that, at first glance,

seems completely unrelated to complex numbers.


Problem 1.1.  Construct a triangle determined by two sides and an angle not included between those sides.

        We will get to Wallis' contributions in a moment. First, observe that Figure 1.1 illustrates the standard solution

to Problem 1.1. Given side length  
[Graphics:Images/ComplexNumberOrigin_gr_1068.gif]  (represented by segment [Graphics:Images/ComplexNumberOrigin_gr_1069.gif]),  angle [Graphics:Images/ComplexNumberOrigin_gr_1070.gif] (determined by segments [Graphics:Images/ComplexNumberOrigin_gr_1071.gif]),

and side length 
[Graphics:Images/ComplexNumberOrigin_gr_1072.gif], draw an arc of a circle of radius [Graphics:Images/ComplexNumberOrigin_gr_1073.gif] whose center is at point [Graphics:Images/ComplexNumberOrigin_gr_1074.gif].  If the arc intersects segment [Graphics:Images/ComplexNumberOrigin_gr_1075.gif] 

at points  
[Graphics:Images/ComplexNumberOrigin_gr_1076.gif],  then the resulting triangles  [Graphics:Images/ComplexNumberOrigin_gr_1077.gif]  each satisfy the problem requirement.

                    [Graphics:Images/ComplexNumberOrigin_gr_1078.gif]  

                              Figure 1.1  The Standard solution to Wallis' geometrical problem.  

 

A Geometric Representation of Real Numbers

The representation of  
[Graphics:Images/ComplexNumberOrigin_gr_1079.gif]  when  [Graphics:Images/ComplexNumberOrigin_gr_1080.gif].

        Wallis' first contribution allowed him to associate (real) numbers with the points [Graphics:Images/ComplexNumberOrigin_gr_1081.gif] of Figure 1.1. The

association came by way of a construct that may sound completely trivial to us, but that is only because we have

been raised knowing about Wallis' idea: "the real number line."  By choosing an arbitrary point to represent the

number zero on a given line, Wallis declared that
positive numbers could be viewed as corresponding distances

to the
right of zero, and negative numbers as corresponding (positive) distances to the left of zero.

        To complete the association refer to Figure 1.2 (a) and think of segment  [Graphics:Images/ComplexNumberOrigin_gr_1082.gif]  as lying on a portion of the

[Graphics:Images/ComplexNumberOrigin_gr_1083.gif]-axis.  Then draw a perpendicular segment  [Graphics:Images/ComplexNumberOrigin_gr_1084.gif],  and designate the origin to be at  [Graphics:Images/ComplexNumberOrigin_gr_1085.gif].  If the length of

[Graphics:Images/ComplexNumberOrigin_gr_1086.gif],  then the Pythagorean theorem gives  [Graphics:Images/ComplexNumberOrigin_gr_1087.gif]  for the length of segments  [Graphics:Images/ComplexNumberOrigin_gr_1088.gif]  and  [Graphics:Images/ComplexNumberOrigin_gr_1089.gif].  Combining

this result with Wallis' number real line results in points  
[Graphics:Images/ComplexNumberOrigin_gr_1090.gif]  representing the real numbers:   

                    
[Graphics:Images/ComplexNumberOrigin_gr_1091.gif],     and     [Graphics:Images/ComplexNumberOrigin_gr_1092.gif].  

                    [Graphics:Images/ComplexNumberOrigin_gr_1093.gif]  

                              Figure 1.2 (a)  Wallis' geometric depiction of "real numbers."  


        For example, if  
[Graphics:Images/ComplexNumberOrigin_gr_1094.gif], the points  [Graphics:Images/ComplexNumberOrigin_gr_1095.gif]  would represent [Graphics:Images/ComplexNumberOrigin_gr_1096.gif], respectively,  because

                    
[Graphics:Images/ComplexNumberOrigin_gr_1097.gif],     and     [Graphics:Images/ComplexNumberOrigin_gr_1098.gif],  

as shown below in the Figure 1.2 (b).

                    [Graphics:Images/ComplexNumberOrigin_gr_1099.gif]  

                    Figure 1.2 (b)  Wallis' geometric depiction of the "real numbers,"   [Graphics:Images/ComplexNumberOrigin_gr_1100.gif].  


     From both an algebraic and geometric viewpoint this procedure only makes sense if the stipulated length [Graphics:Images/ComplexNumberOrigin_gr_1101.gif]

is greater than or equal to 
[Graphics:Images/ComplexNumberOrigin_gr_1102.gif].  If  [Graphics:Images/ComplexNumberOrigin_gr_1103.gif]  were less than  [Graphics:Images/ComplexNumberOrigin_gr_1104.gif]  then the algebraic expressions for points [Graphics:Images/ComplexNumberOrigin_gr_1105.gif]

and  
[Graphics:Images/ComplexNumberOrigin_gr_1106.gif]  would be meaningless, as the quantity  [Graphics:Images/ComplexNumberOrigin_gr_1107.gif]  inside the square root would be negative.

Viewed geometrically,  if  
[Graphics:Images/ComplexNumberOrigin_gr_1108.gif]  was less than  [Graphics:Images/ComplexNumberOrigin_gr_1109.gif]  then the arc of radius  [Graphics:Images/ComplexNumberOrigin_gr_1110.gif]  that is centered at  [Graphics:Images/ComplexNumberOrigin_gr_1111.gif]  would not be

able to intersect the segment  [Graphics:Images/ComplexNumberOrigin_gr_1112.gif].  In other words, if  
[Graphics:Images/ComplexNumberOrigin_gr_1113.gif]  were less than  [Graphics:Images/ComplexNumberOrigin_gr_1114.gif]  Problem 1.1 would appear

to have no solution.  


A Geometric Representation of Complex Numbers

The representation of  
[Graphics:Images/ComplexNumberOrigin_gr_1115.gif]  when  [Graphics:Images/ComplexNumberOrigin_gr_1116.gif].

        Appearances, of course,  can be deceiving, and Wallis reinforced the truth of that ancient proverb when

he came up with his second contribution.  It was a solution to Problem 1.1 in the case when  
[Graphics:Images/ComplexNumberOrigin_gr_1117.gif]  is less than  [Graphics:Images/ComplexNumberOrigin_gr_1118.gif].  

Figure 1.3 (a) illustrates how he did it.  From the midpoint of 
[Graphics:Images/ComplexNumberOrigin_gr_1119.gif] Wallis drew a circle with diameter [Graphics:Images/ComplexNumberOrigin_gr_1120.gif]. Then,

with  
[Graphics:Images/ComplexNumberOrigin_gr_1121.gif]  as a center he drew an arc of radius  [Graphics:Images/ComplexNumberOrigin_gr_1122.gif].  Because  [Graphics:Images/ComplexNumberOrigin_gr_1123.gif]  is less than  [Graphics:Images/ComplexNumberOrigin_gr_1124.gif]  the arc will intersect the circle

at two points, say  
[Graphics:Images/ComplexNumberOrigin_gr_1125.gif],  which now lie above the "real number line."  

                    [Graphics:Images/ComplexNumberOrigin_gr_1126.gif]  

                    Figure 1.3 (a).  Wallis' geometric depiction of the "complex numbers,"   [Graphics:Images/ComplexNumberOrigin_gr_1127.gif].


        Again we get two triangles: [Graphics:Images/ComplexNumberOrigin_gr_1128.gif]. Wallis claimed that these triangles each satisfy the requirement

of Problem 1.1.  You might object to this construction on the grounds that angle  
[Graphics:Images/ComplexNumberOrigin_gr_1129.gif]  is not part of either triangle.

However, if you read the problem statement carefully, you will notice that it never states that the angle
[Graphics:Images/ComplexNumberOrigin_gr_1130.gif] had

to be part of any triangle, only that it determine a triangle.  From this perspective Wallis completely satisfied

the requirement.

        Notice that the points  [Graphics:Images/ComplexNumberOrigin_gr_1131.gif]  are no longer on the [Graphics:Images/ComplexNumberOrigin_gr_1132.gif]-axis as they were when  [Graphics:Images/ComplexNumberOrigin_gr_1133.gif]  was greater than  [Graphics:Images/ComplexNumberOrigin_gr_1134.gif],

(and  
[Graphics:Images/ComplexNumberOrigin_gr_1135.gif]are real numbers). They are now located somewhere above the [Graphics:Images/ComplexNumberOrigin_gr_1136.gif]-axis, and it is reasonable

to conclude that
[Graphics:Images/ComplexNumberOrigin_gr_1137.gif] give the geometric representations of the expressions [Graphics:Images/ComplexNumberOrigin_gr_1138.gif]

when
[Graphics:Images/ComplexNumberOrigin_gr_1139.gif], (and  [Graphics:Images/ComplexNumberOrigin_gr_1140.gif] are complex numbers).

     Although Wallis only hinted at such a conclusion, he nevertheless helped set the stage for thinking about real

numbers as being embedded in a larger set of complex numbers, and that these numbers could be represented

geometrically as vertical displacements from the
[Graphics:Images/ComplexNumberOrigin_gr_1141.gif]-axis.  Unfortunately, if we tried to apply Wallis' method to

construct complex numbers we would find it had some serious defects.

 

Wallis' depiction of  [Graphics:Images/ComplexNumberOrigin_gr_1142.gif]  

                    [Graphics:Images/ComplexNumberOrigin_gr_1143.gif]  

                    Figure 1.3 (b).  An interpretation of Wallis' geometric depiction of   "[Graphics:Images/ComplexNumberOrigin_gr_1144.gif]   and   [Graphics:Images/ComplexNumberOrigin_gr_1145.gif]".


 Suppose that we choose [Graphics:Images/ComplexNumberOrigin_gr_1146.gif], then Wallis' expression [Graphics:Images/ComplexNumberOrigin_gr_1147.gif] becomes [Graphics:Images/ComplexNumberOrigin_gr_1148.gif], and points [Graphics:Images/ComplexNumberOrigin_gr_1149.gif]

now coincide at point  
[Graphics:Images/ComplexNumberOrigin_gr_1150.gif], as shown above in Figure 1.3 (b).  For certain, we can say that the value  "[Graphics:Images/ComplexNumberOrigin_gr_1151.gif]"

is the proper choice we use today.  So in some sense, Wallis was the first person to place the point  [Graphics:Images/ComplexNumberOrigin_gr_1152.gif]  in

it's proper location in the upper half-plane.  But we surely would not want to equate the points  "[Graphics:Images/ComplexNumberOrigin_gr_1153.gif]  and

[Graphics:Images/ComplexNumberOrigin_gr_1154.gif]."  The reason for this anomaly was that the entire complex plane had not yet been invented.  Thus,

even with Wallis' work the jigsaw of getting the proper picture of complex numbers remained.  It would be

another century before someone put most of the pieces together.

 

Caspar Wessel Makes a Breakthrough

     The eighteenth century Norwegian mathematician Caspar Wessel (1745-1818) made the next contribution.

On
March 10, 1797 Caspar Wessel presented a paper to the Danish Academy of Sciences entitled "On the

Analytic Representation of Direction: An Attempt."  In that paper he described how to manipulate vectors

geometrically, and this description eventually led to the current geometric representation of complex numbers

that we use today.    

        Simply put, vectors are directed line segments.  To add two vectors make a copy of the second vector

and place its tail onto the head of the first vector. The resultant vector is the directed line segment drawn from

the tail of the first vector to the head of the second copy vector.  Figure 1.4 (a) illustrates the addition of vector

[Graphics:Images/ComplexNumberOrigin_gr_1155.gif]  to vector  [Graphics:Images/ComplexNumberOrigin_gr_1156.gif].

        The procedure of adding vectors had been well-known for some time. The unique contribution that Wessel

made was his description of how to multiply two vectors.  

        To understand Wessel's thinking recall that any vector can be represented by two quantities: its length, and

its angular displacement from the positive [Graphics:Images/ComplexNumberOrigin_gr_1157.gif]-axis. Figure 1.4 (b) illustrates this idea for the vector labeled as [Graphics:Images/ComplexNumberOrigin_gr_1158.gif]:

it's length is  [Graphics:Images/ComplexNumberOrigin_gr_1159.gif],  and its angular displacement from the positive [Graphics:Images/ComplexNumberOrigin_gr_1160.gif]-axis is  [Graphics:Images/ComplexNumberOrigin_gr_1161.gif].

[Graphics:Images/ComplexNumberOrigin_gr_1162.gif][Graphics:Images/ComplexNumberOrigin_gr_1163.gif]

                          (a)  Addition of two vectors                                                 (b)  Length and direction of a vector  

                                                  Figure 1.4  Wessel's construction of for the geometry of vectors.


     Wessel stated that, to multiply two vectors, the length of the product vector should simply be the product

of the lengths of its factors.  Should the
angular displacement of the product vector then be the product of the

angular displacements its factors?  Definitely not, and you will see in the exercises why such a provision would

be a bad idea.  What then, should be the
angular displacement of the product?  

     In answering this question Wessel drew an analogy from the multiplication of real numbers. He observed that,

                    if    
[Graphics:Images/ComplexNumberOrigin_gr_1164.gif],    then    [Graphics:Images/ComplexNumberOrigin_gr_1165.gif]    and    [Graphics:Images/ComplexNumberOrigin_gr_1166.gif].  

In other words, the ratio of the product to any of its factors is the same as the ratio other factor to the number one.

       What vector represents the number one?  It seems obvious that, using the number line of Wallis, it should be

the directed line segment between the origin and the number "one" on the positive
[Graphics:Images/ComplexNumberOrigin_gr_1167.gif]-axis.  

Let's call this vector the standard unit vector,
[Graphics:Images/ComplexNumberOrigin_gr_1168.gif] , as illustrated in Figure 1.5.

                                    [Graphics:Images/ComplexNumberOrigin_gr_1169.gif]  

                              Figure 1.5  Wessel's view of the standard unit vector  [Graphics:Images/ComplexNumberOrigin_gr_1170.gif].


      With this identification in mind (and using the multiplication analogy just mentioned) Wessel made a brilliant move.

He saw that the
angular displacement of the product of two vectors should differ from the angular displacement

of each factor by the same amount that the
angular displacement of the other factor differs from the angular

displacement of the standard unit vector.  That's quite a mouthful;  let's see what it means.

     What is the
angular displacement of the standard unit vector  [Graphics:Images/ComplexNumberOrigin_gr_1171.gif] ?  Clearly, its angular displacement is zero

radians, as it coincides with the positive
[Graphics:Images/ComplexNumberOrigin_gr_1172.gif]-axis. Thus, if vectors [Graphics:Images/ComplexNumberOrigin_gr_1173.gif] and [Graphics:Images/ComplexNumberOrigin_gr_1174.gif] have angular displacements of [Graphics:Images/ComplexNumberOrigin_gr_1175.gif],

respectively, and vector
 [Graphics:Images/ComplexNumberOrigin_gr_1176.gif], then the angular displacement of [Graphics:Images/ComplexNumberOrigin_gr_1177.gif] should be [Graphics:Images/ComplexNumberOrigin_gr_1178.gif], as seen in Figure 1.6 (a).

 The reason for this is that, with such an arrangement, Wessel's displacement protocol works out perfectly: the

displacement of
  [Graphics:Images/ComplexNumberOrigin_gr_1179.gif]  (which is  [Graphics:Images/ComplexNumberOrigin_gr_1180.gif]) differs from the displacement of  [Graphics:Images/ComplexNumberOrigin_gr_1181.gif]  (which is  [Graphics:Images/ComplexNumberOrigin_gr_1182.gif])  by  [Graphics:Images/ComplexNumberOrigin_gr_1183.gif].  This is the same

amount that the
angular displacement of  [Graphics:Images/ComplexNumberOrigin_gr_1184.gif] (which is[Graphics:Images/ComplexNumberOrigin_gr_1185.gif]) differs from the angular displacement of the standard

unit vector
 [Graphics:Images/ComplexNumberOrigin_gr_1186.gif] (which is [Graphics:Images/ComplexNumberOrigin_gr_1187.gif]). The angular displacement of  [Graphics:Images/ComplexNumberOrigin_gr_1188.gif] differs from the angular displacement of [Graphics:Images/ComplexNumberOrigin_gr_1189.gif] by [Graphics:Images/ComplexNumberOrigin_gr_1190.gif],

which is the same amount that the
angular displacement of  [Graphics:Images/ComplexNumberOrigin_gr_1191.gif]  differs from the angular displacement of the

standard unit vector
  [Graphics:Images/ComplexNumberOrigin_gr_1192.gif].  

[Graphics:Images/ComplexNumberOrigin_gr_1193.gif][Graphics:Images/ComplexNumberOrigin_gr_1194.gif]

               (a)  Multiplication of two vectors.                                                              (b)  The square root of  [Graphics:Images/ComplexNumberOrigin_gr_1195.gif].

                                                         Figure 1.6  Wessel's multiplication scheme for vectors.  

 

Wessel's vector  [Graphics:Images/ComplexNumberOrigin_gr_1196.gif]  

     How does Wessel's procedure lead to a geometric representation of complex numbers? Consider what happens

if a unit vector is drawn from the origin straight up the
[Graphics:Images/ComplexNumberOrigin_gr_1197.gif]-axis, and then multiplied by itself.  By Wessel's rules the

length of the product vector is one unit, as the length of each factor is one unit.  What about its direction?  The

angular displacement of the original vector is  [Graphics:Images/ComplexNumberOrigin_gr_1198.gif]  radians, so by Wessel's rules again the product vector has an

angular displacement of  [Graphics:Images/ComplexNumberOrigin_gr_1199.gif] radians.  Thus, the product vector is aligned along the [Graphics:Images/ComplexNumberOrigin_gr_1200.gif]-axis, but is

directed from the origin to
the left by one unit, as shown in Figure 1.6 (b). Using Wallis' number line we see

that the product vector is naturally identified with the number  
[Graphics:Images/ComplexNumberOrigin_gr_1201.gif].  Label the original vector as  [Graphics:Images/ComplexNumberOrigin_gr_1202.gif].  

What do you conclude?  Obviously, that  
[Graphics:Images/ComplexNumberOrigin_gr_1203.gif],  which must mean that  [Graphics:Images/ComplexNumberOrigin_gr_1204.gif]. Neat!  

        Neat, yes, but the material we presented leading up to this result was (if you'll pardon the pun) complex.  

Thus, you need not worry if you had some difficulty following it.  Sections 1.2-1.5 will flesh out these ideas in

much more detail, and we will be pleased to find that the term
angular displacement is replaced with "argument,"

and complex numbers will be easier to understand.  Another famous result is attributed to the French

mathematician Abraham de Moivre (1667-1754) is remembered for his formula

                    [Graphics:Images/ComplexNumberOrigin_gr_1205.gif],  

which took trigonometry into complex analysis (see Section 1.5).

        It should be pointed out that Wessel was not the only mathematician - or even the first - who began thinking

of complex numbers as vectors, and points in the plane." As early as
1732 the Swiss mathematician Leonard Euler

(1707-1783) adopted this view concerning the [Graphics:Images/ComplexNumberOrigin_gr_1206.gif] solutions to the equation [Graphics:Images/ComplexNumberOrigin_gr_1207.gif]. In Section 1.5 we show that

these solutions can be expressed
[Graphics:Images/ComplexNumberOrigin_gr_1208.gif]  for certain values of  [Graphics:Images/ComplexNumberOrigin_gr_1209.gif].  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
[Graphics:Images/ComplexNumberOrigin_gr_1210.gif]  for  [Graphics:Images/ComplexNumberOrigin_gr_1211.gif].

Today this notation is still the most popular, although some electrical engineers prefer the symbol [Graphics:Images/ComplexNumberOrigin_gr_1212.gif] instead,

so that they can use
[Graphics:Images/ComplexNumberOrigin_gr_1213.gif] to represent current.

    Is it possible to modify slightly Wallis's picture of complex numbers so 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.

     In the nineteenth century there were many contributions. The French mathematician Augustin Louis Cauchy

(1789-1857) contributed theorems that are part of the body of complex analysis.  The German mathematician

 Johann Carl Friedrich Gauss (1777-1855) reinforced the utility of complex numbers by using them in several

proofs of the Fundamental Theorem of Algebra, (see Section 6.6).  In an 1831 paper, he produced a clear

geometric representation of  [Graphics:Images/ComplexNumberOrigin_gr_1214.gif]  by identifying it with the point [Graphics:Images/ComplexNumberOrigin_gr_1215.gif] in the coordinate plane.  He also

described the arithmetic operations with these new complex 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 (1806-1871) commented in his book, On the Study

and Difficulties of Mathematics,
"We have shown the symbol  [Graphics:Images/ComplexNumberOrigin_gr_1216.gif]  to be void of meaning, or rather self-

contradictory and absurd.  By means of such symbols, a part of algebra is established which is of great utility."

    There are genuine logical problems associated with complex numbers.  For example, with real numbers 

[Graphics:Images/ComplexNumberOrigin_gr_1217.gif]  if both sides of the equation are defined. Applying this identity to complex numbers leads to

                    [Graphics:Images/ComplexNumberOrigin_gr_1218.gif].   

Plausible answers to these problems can be given, however, and you will learn how to resolve this apparent

contradiction in Section 2.2. De Morgan's remark illustrates that many factors are needed in order to persuade

mathematicians to adopt new theories.  In this case, as always, a logical foundation was crucial, but so too was

a willingness to modify some ideas concerning certain well-established properties of numbers.

     Another German mathematician Georg Friedrich Bernhard Riemann (1826-1866), made many contributions

in particular he is accredited with inventing Riemann surfaces, which we will discuss many times in the book

(See Sections 2.4, 5.2, 10.3, etc.).

    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.

 

Relevance for Today's Computer Software

        One might wonder what utility there could be for the "Ferro-Tartaglia" formula that was invented almost 500

years ago.
 If you use [Graphics:Images/ComplexNumberOrigin_gr_1219.gif], you will find that a solution to the depressed cubic  [Graphics:Images/ComplexNumberOrigin_gr_1220.gif]  is either

          [Graphics:Images/ComplexNumberOrigin_gr_1221.gif]     

or

     [Graphics:Images/ComplexNumberOrigin_gr_1222.gif],

depending on whether you used an older version Mathematica 4, or the current version Mathematica 7,
          
and [Graphics:Images/ComplexNumberOrigin_gr_1223.gif] will find that the solution   [Graphics:Images/ComplexNumberOrigin_gr_1224.gif].  

We leave it for the reader to verify that all three of these expressions are algebraically equivalent to the following

                    [Graphics:Images/ComplexNumberOrigin_gr_1225.gif].  

Simplifying the first term in the expression for  [Graphics:Images/ComplexNumberOrigin_gr_1226.gif]  we have  

                    [Graphics:Images/ComplexNumberOrigin_gr_1227.gif]  

Simplifying the second term in the expression for  [Graphics:Images/ComplexNumberOrigin_gr_1228.gif]  we have  

     [Graphics:Images/ComplexNumberOrigin_gr_1229.gif]  

Therefore,

                     [Graphics:Images/ComplexNumberOrigin_gr_1230.gif]

          This might be a welcome conclusion and might lead you to use Ferro-Tartaglia formula with computer

algebra software.  But recall that complex numbers had not yet been invented in 1545, and the Ferro-Tartaglia

formula formula was used only to find a real root.  Should we be careful when using this formula?  In this book

we will learn that there are two serious errors in the above alleged algebraic simplification.

        The first error involves the identity  [Graphics:Images/ComplexNumberOrigin_gr_1231.gif]  which is not true for complex numbers.  If you use a

computer algebra system to compute [Graphics:Images/ComplexNumberOrigin_gr_1232.gif] then the principal value is computed. We will learn about cube

roots in Section 1.5, and the principal value of the cube root in Section 2.2.  

        The second error involves the principal value of the square roots and principal value of the cube roots.  

The following identity is not valid  for some choices of  [Graphics:Images/ComplexNumberOrigin_gr_1233.gif],   

                    [Graphics:Images/ComplexNumberOrigin_gr_1234.gif].  

        However, there seems to be the following serendipity for the Ferro-Tartaglia formulae.


Conjecture. (The Ferro-Tartaglia Formulae).  If  [Graphics:Images/ComplexNumberOrigin_gr_1235.gif] and [Graphics:Images/ComplexNumberOrigin_gr_1236.gif] are real or complex constants used to form

the depressed cubic equation

                    [Graphics:Images/ComplexNumberOrigin_gr_1237.gif],  

then one (and sometimes two) of the following four calculations will be a root (possibly a complex root).  

                    [Graphics:Images/ComplexNumberOrigin_gr_1238.gif]  

Remark.  The above calculations are to be done in complex arithmetic with Mathematica or Maple using the

principal values of the square roots and cube roots.  We applaud Tartaglia, Ferro, Cardano, and Bombelli,

for introducing a formula which can be tweaked to produce a root of any depressed cubic equation.  We do

not have a proof for the conjecture, let us know if you come up with one.  Otherwise it is advisable to use the

"solve" subroutines that are included in the software programs.


Example 8.   Given Cardano's depressed cubic equation

                    [Graphics:Images/ComplexNumberOrigin_gr_1239.gif].  

Use Mathematica and Maple and the Ferro-Tartaglia formulae to calculate the four values

[Graphics:Images/ComplexNumberOrigin_gr_1240.gif] in the conjecture, and then determine which one is a solution to the cubic.

Explore Example 8.

 

Example 9.   Given the cubic equation

                    [Graphics:Images/ComplexNumberOrigin_gr_1287.gif].  

Then the corresponding depressed cubic equation is

                    [Graphics:Images/ComplexNumberOrigin_gr_1288.gif].  

Use Mathematica and Maple and the Ferro-Tartaglia formulae to calculate the four values

[Graphics:Images/ComplexNumberOrigin_gr_1289.gif] in the conjecture, and then determine ones are solutions to the cubic.

Explore Example 9.

 

 

The solution of quadratic equations.  

[Graphics:Images/ComplexNumberOrigin_gr_1420.gif] and  [Graphics:Images/ComplexNumberOrigin_gr_1421.gif] can construct the solutions to the general quadratic equation.

Quadratic Formula Exploration.

 

The solution of cubic equations.  

[Graphics:Images/ComplexNumberOrigin_gr_1431.gif] and  [Graphics:Images/ComplexNumberOrigin_gr_1432.gif] can construct the solutions to the general cubic equation.  

Cubic Formula Exploration.

 

 

     The Ferro-Tartaglia conjecture was probably known, but not explicitly stated.  Because it relies on the special

trick that we teach in algebra, namely that  [Graphics:Images/ComplexNumberOrigin_gr_1461.gif]  when  r is a positive real number.  This seems to be a

remnant of doing arithmetic in the "real domain" and in Section 1.5 we will learn that there are three complex

cube roots and in Section 2.2 we will learn that there are branches of  the square root and cube root functions

which must be used in complex computations.  

 

Theorem. (The Ferro-Tartaglia-Cardano Formulae).  If  [Graphics:Images/ComplexNumberOrigin_gr_1462.gif] and [Graphics:Images/ComplexNumberOrigin_gr_1463.gif] are real or complex constants

used to form the depressed cubic equation

                    [Graphics:Images/ComplexNumberOrigin_gr_1464.gif],  

then all three roots will appear in the list of nine calculations,    

                    [Graphics:Images/ComplexNumberOrigin_gr_1465.gif] ,

for  [Graphics:Images/ComplexNumberOrigin_gr_1466.gif]  and  [Graphics:Images/ComplexNumberOrigin_gr_1467.gif] ,  

where   [Graphics:Images/ComplexNumberOrigin_gr_1468.gif],   [Graphics:Images/ComplexNumberOrigin_gr_1469.gif]   and   [Graphics:Images/ComplexNumberOrigin_gr_1470.gif].  

Remark.  This theorem might seem awkward to be of practical use, because it would require one to test each

of the nine tentative solutions by actually calculating  [Graphics:Images/ComplexNumberOrigin_gr_1471.gif] for    and  for  [Graphics:Images/ComplexNumberOrigin_gr_1472.gif]  and  [Graphics:Images/ComplexNumberOrigin_gr_1473.gif],

and then choosing which three are the desired distinct solutions.  For that reason we omit its proof.  However,

we will see in Section 2.2 that Vieta's substitution will reduce the number of tentative solutions to six guesses.

After we introduce the principle cube root of unity in Section 1.5, it is easy to show how to choose the correct

three distinct complex roots for the cubic equation.  

 

Example 10.  Given Cardano's depressed cubic equation

                  [Graphics:Images/ComplexNumberOrigin_gr_1474.gif].  

Use Mathematica to calculate the nine values  [Graphics:Images/ComplexNumberOrigin_gr_1475.gif] with the Ferro-Tartaglia-Cardano formulae,

and then determine which three values are solutions to the cubic equation.

Explore Example 10.

 

Example 11.  Given Cardano's depressed cubic equation

                  [Graphics:Images/ComplexNumberOrigin_gr_1537.gif].  

Use Mathematica to calculate the nine values  [Graphics:Images/ComplexNumberOrigin_gr_1538.gif] with the Ferro-Tartaglia-Cardano formulae,

and then determine which three values are solutions to the cubic equation.

Explore Example 11.

 

 

The Cube Root Fallacy  

          Does   [Graphics:Images/ComplexNumberOrigin_gr_1600.gif]   imply that   [Graphics:Images/ComplexNumberOrigin_gr_1601.gif] ?       

Sometimes we try to do computations rapidly, without thinking.  For example we might try to solve

the above equation by extracting the cube root of both sides and writing

                    [Graphics:Images/ComplexNumberOrigin_gr_1602.gif],  

and then make the simplification

                    [Graphics:Images/ComplexNumberOrigin_gr_1603.gif],  

and with a sigh of "anxiety" obtain

                    [Graphics:Images/ComplexNumberOrigin_gr_1604.gif].  

Remark.  This paradox can is resolved by considering the cube roots of unity, which are discussed in Section 1.5.

Then we will be able to determine that there are two solutions   [Graphics:Images/ComplexNumberOrigin_gr_1605.gif] .  

The details are also discussed in the article  "The Cube Root Fallacy: Does [Graphics:Images/ComplexNumberOrigin_gr_1606.gif] Imply that [Graphics:Images/ComplexNumberOrigin_gr_1607.gif] ?",

The AMATYC Review, Vol., 24, No. 2, Spring, 2003, pp 77-79.  

 

Graphing the ordinary cube root function.  

        Did your algebra or calculus course include graphing ?  Perhaps you used a graphics calculator for the task.

Then you might feel comfortable with the following graph of  [Graphics:Images/ComplexNumberOrigin_gr_1608.gif]  over the interval  [Graphics:Images/ComplexNumberOrigin_gr_1609.gif].  

                                   [Graphics:Images/ComplexNumberOrigin_gr_1610.gif]    

                    Graph of  [Graphics:Images/ComplexNumberOrigin_gr_1611.gif],  over the interval  [Graphics:Images/ComplexNumberOrigin_gr_1612.gif],  as taught in algebra and calculus.

        Have you used [Graphics:Images/ComplexNumberOrigin_gr_1613.gif] or [Graphics:Images/ComplexNumberOrigin_gr_1614.gif] or [Graphics:Images/ComplexNumberOrigin_gr_1615.gif] to graph   [Graphics:Images/ComplexNumberOrigin_gr_1616.gif]  over the interval  [Graphics:Images/ComplexNumberOrigin_gr_1617.gif] ?

If you have not used computer software then what you are about to see might seem out of the ordinary.

        The [Graphics:Images/ComplexNumberOrigin_gr_1618.gif] command to draw the graph of  [Graphics:Images/ComplexNumberOrigin_gr_1619.gif]  is

        [Graphics:Images/ComplexNumberOrigin_gr_1620.gif]

        The [Graphics:Images/ComplexNumberOrigin_gr_1621.gif] command  to draw the graph  [Graphics:Images/ComplexNumberOrigin_gr_1622.gif]  is similar  

       [Graphics:Images/ComplexNumberOrigin_gr_1623.gif]

The resulting graph for  [Graphics:Images/ComplexNumberOrigin_gr_1624.gif]  that is drawn is is given below.

                              [Graphics:Images/ComplexNumberOrigin_gr_1626.gif]  

                    Graph of  [Graphics:Images/ComplexNumberOrigin_gr_1627.gif],  over the interval  [Graphics:Images/ComplexNumberOrigin_gr_1628.gif],  as drawn with [Graphics:Images/ComplexNumberOrigin_gr_1629.gif] or [Graphics:Images/ComplexNumberOrigin_gr_1630.gif].


     Why isn't the graph drawn when  [Graphics:Images/ComplexNumberOrigin_gr_1631.gif] ?  Why doesn't it look like the one from algebra and calculus ?

The answer is easy. All three  computer algebra programs: [Graphics:Images/ComplexNumberOrigin_gr_1632.gif], [Graphics:Images/ComplexNumberOrigin_gr_1633.gif] and [Graphics:Images/ComplexNumberOrigin_gr_1634.gif] use complex

number arithmetic in all computations, including graphics. The cube root of negative numbers are computed as

the principle complex cube root which involves an imaginary component.  Thus, when a complex number is

computed for  [Graphics:Images/ComplexNumberOrigin_gr_1635.gif] over the interval [Graphics:Images/ComplexNumberOrigin_gr_1636.gif]  it cannot be plotted, and [Graphics:Images/ComplexNumberOrigin_gr_1637.gif] and [Graphics:Images/ComplexNumberOrigin_gr_1638.gif]

make the graph blank.

     The situation is more bizarre if you use the computer software [Graphics:Images/ComplexNumberOrigin_gr_1639.gif].  

The [Graphics:Images/ComplexNumberOrigin_gr_1640.gif] commands to graph of  [Graphics:Images/ComplexNumberOrigin_gr_1641.gif]  are

          [Graphics:Images/ComplexNumberOrigin_gr_1642.gif]

First, [Graphics:Images/ComplexNumberOrigin_gr_1643.gif] will print the following error warning message.  

          [Graphics:Images/ComplexNumberOrigin_gr_1644.gif]

Then, [Graphics:Images/ComplexNumberOrigin_gr_1645.gif] will draw the following mysterious graph for  [Graphics:Images/ComplexNumberOrigin_gr_1646.gif]  over the entire interval  [Graphics:Images/ComplexNumberOrigin_gr_1647.gif],  

                              [Graphics:Images/ComplexNumberOrigin_gr_1648.gif]  

                                        Graph of  [Graphics:Images/ComplexNumberOrigin_gr_1649.gif],  as drawn with [Graphics:Images/ComplexNumberOrigin_gr_1650.gif].  

Can you tell what is happening here with the [Graphics:Images/ComplexNumberOrigin_gr_1651.gif] graph ?  Why is the graph positive when  [Graphics:Images/ComplexNumberOrigin_gr_1652.gif] ?

The answer is easy.  [Graphics:Images/ComplexNumberOrigin_gr_1653.gif] is actually graphing  [Graphics:Images/ComplexNumberOrigin_gr_1654.gif],  where [Graphics:Images/ComplexNumberOrigin_gr_1655.gif] is the real part

of the complex number  [Graphics:Images/ComplexNumberOrigin_gr_1656.gif].  

 

Concluding Remarks.  

     If you plan to use computers to do analysis then be prepared to learn complex analysis. Because, the

three major software programs; [Graphics:Images/ComplexNumberOrigin_gr_1657.gif], [Graphics:Images/ComplexNumberOrigin_gr_1658.gif] and [Graphics:Images/ComplexNumberOrigin_gr_1659.gif] all do their calculations using complex

number arithmetic.  Sometimes we get surprises when we least expect them.  Our goal is to find out how

complex analysis can be used properly. We will learn about the real part of a complex numbers in Section 1.2,

and the principal value of the cube root in Section 2.2.  There is plenty of excitement ahead.  

 

Acknowledgement  

     This Maple worksheet is a complementary supplement to accompany our textbook Complex Analysis

for Mathematics and Engineering,   Jones and Bartlett Learning.  We appreciate receiving correspondence

regarding the textbook and worksheets.  You are welcome to correspond with us by mail or e-mail.

                    Prof. John  H.  Mathews  
                    Department of Mathematics  
                    California State University Fullerton  
                    Fullerton, CA  92634  
                    mathews@fullerton.edu  

                    Prof. Russell W. Howell  
                    Mathematics & Computer Science Department  
                    Westmont  College  
                    Santa Barbara,  CA  93108  
                    howell@westmont.edu  

 

Exercises for Section 1.1.  The Origin of Complex Numbers

 

 

Library Research Experience for Undergraduates

History of Complex Numbers

Complex Numbers

 Fundamental Theorem of Algebra

 

Download this Mathematica Notebook

 Download the Maple Worksheet

 

 

The Next Module is

Complex Number Algebra

 

 

Return to the Complex Analysis Modules  

 

 

Return to the Complex Analysis Project

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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