**for**

**Chapter 1 Complex
Numbers**

**Preliminary Remarks**

Complex numbers
are introduced in algebra courses and the
letter is
used to denote , and
it is called

the imaginary unit. Since , it
is the solution of the equation . Then
the notation 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

,

the roots are

.

Then cubic
equations are introduced. Methods for solving a cubic
equation require one real root
to be given

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

,

the hint that the real root is
might be given, or it might be left the student to discover.
Then next step is

to factor the cubic as

.

To solve the equation , the
method of factoring can be used or the
quadratic formula can be
used.

The solutions to

are easily found to be .

Then the three solutions to are
given by

.

Aside. We can use
*Mathematica* or Maple to plot the roots
of . This
is just for fun!

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

Graphs
of the cubic polynomial .

Can
you locate the roots
?

**Remarks.** We
will introduce the notation in
Section
1.2.

We will discuss , the
absolute value (or modulus) in Section
1.3,

and the argument
in Section
1.4.

We will introduce, , a
function of the complex variable , in
Section
2.1.

The solution to
the general cubic equation 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 ." 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."

To prepare our thinking, let us
focus on the following special way to
solve .

Since the equation is a quadratic, we must have , and
we can divide each term by and
get,

.

Now use the special
substitution and
get,

.

which can be simplified to obtain the depressed
quadratic equation,

.

It is an easy task solve this depressed
quadratic equation, because there is no linear
term ,

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

.

then take the square root and obtain the two
solutions,

,

and

.

Then recall the special substitution
that we used ,

construct the two solutions to the given quadratic
equation ,

,

and

,

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

.

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

.

If you make the substitution , this
will transform the general cubic equation into

,

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

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

**Exploration
for Cardano's Substitution.**

**Cardano's Example
1.** Given the cubic equation

.

If you make the substitution , this
will transform the cubic equation into

,

which simplifies to

.

**Exploration
for Cardano's Example 1.**

**The
Ferro-Tartaglia-Cardano Formula**

If Cardano could get any value
that solved a depressed cubic, he could easily construct a solution
to

,

by using the substitution . 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 to
the depressed cubic.

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

,

is

.

**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
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 and

use it to factor the depressed cubic into a linear and quadratic
term. The remaining two roots, ,

can then be found with the quadratic formula.

**Cardano's Example
2.** Solve the cubic
equation

.

**Cardano's Solution
2.**

To solve , notice
that
and use the substitution and
get

Hence,

,

is the corresponding depressed cubic
equation.

Next, apply the "Ferro-Tartaglia" formula
with and and
calculate

Hence,

is a root.

Next, divide into and
get

Thus,

.

Hence, is
a factor of the depressed cubic and
we have

.

Now it is easy to see that the remaining (duplicate) roots
of are .

Therefore, the solutions to

,

are obtained by recalling the substitution , which
yields

**Exploration
for Cardano's Example 2.**

**Example
3.** Given depressed cubic
equation

.

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

**Example
4.** Given depressed cubic
equation

.

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

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

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.

**Bombelli's Example
5.** Solve the depressed cubic
equation

.

**Bombelli's Solution
5.**

Bombelli then used the "Ferro-Tartaglia" formula for finding a
solution to , which
is

.

To solve , he
substituted and and
computed

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

in the preceding equation could be written as

and

for some real numbers . That
is, Bombelli believed

and ,

which would mean

and .

Then, using the well-known algebraic identity , (letting and ),

and assuming that roots of negative numbers obey the rules of
algebra, he expanded and
obtained

By equating like
parts, in the last two lines, ,

Bombelli reasoned that

,

and

.

Perhaps thinking even more wildly, Bombelli then supposed that
should be integers. The only integer

factors of , so
in the first equation, , Bombelli
concluded that

and .

Using his conclusion,
becomes
which simplifies as .
Then it follows that

and ,

and there are two choices . Amazingly,
he also found that are
solutions to the second equation

,

so Bombelli declared that the values for should
be , respectively.

Using these values in his
equation , he
could now declare that

,

and then he could take the cube root and write

.

By a similar line of reasoning, it follows that

.

Then, this implies that

,

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

of the symbol "" 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
to the depressed cubic equation,

.

**Example
7.** Use Bombelli's method to find the
integer solution
to the depressed cubic equation,

.

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 was
a correct

solution to the equation 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.
**

** **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 *
",

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 * *(represented
by segment ), angle * *(determined
by segments ),

and side length , draw
an arc of a circle of radius whose
center is at point . If
the arc intersects segment *
*at points , then
the resulting triangles

**Figure
1.1** The Standard
solution to Wallis' geometrical problem.

**A Geometric
Representation of Real Numbers
The representation
of **

Wallis'
first contribution allowed him to associate (real) numbers with the
points * *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 * *as
lying on a portion of the

-axis. Then
draw a perpendicular segment ,* *and
designate the origin to be at . If
the length of

*,* then
the Pythagorean theorem gives for
the length of segments * *and .* *Combining

this result with Wallis' number real line results in
points representing
the real numbers:

, and .

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

For example,
if ,
the points * *would
represent , respectively, because

, and ,

as shown below in the Figure 1.2 (b).

**Figure
1.2 (b)** Wallis'
geometric depiction of the "real numbers,"
.

From both
an algebraic and geometric viewpoint this procedure only makes sense
if the stipulated length

is greater than or equal to . If were
less than then
the algebraic expressions for points

and would
be meaningless, as the quantity inside
the square root would be negative.

Viewed geometrically, if was
less than then
the arc of radius that
is centered at would
not be

able to intersect the segment . In
other words, if were
less than Problem
1.1 would appear

to have no solution.

**A Geometric
Representation of Complex Numbers
The representation
of **

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 is less than .

Figure 1.3 (a) illustrates how he did it. From the midpoint of Wallis drew a circle with diameter . Then,

with as a center he drew an arc of radius . Because is less than the arc will intersect the circle

at two points, say , which now lie above the "

**Figure
1.3 (a).** Wallis' geometric
depiction of the "complex numbers," .

Again
we get two triangles: .
Wallis claimed that these triangles each satisfy the requirement

of Problem 1.1. You might object to this construction on
the grounds that angle * *is
not part of either triangle.

However, if you read the problem statement carefully, you will notice
that it never states that the angle *
*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 are
no longer on the -axis
as they were when * *was
greater than ,*
*(and are
real numbers). They are now located somewhere above the
-axis,
and it is reasonable

to conclude that give the geometric representations of the expressions

when

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 -axis. Unfortunately, if we tried to apply Wallis' method to

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

**Wallis' depiction
of **

**Figure
1.3 (b).** An interpretation of Wallis'
geometric depiction
of " and ".

Suppose that we
choose , then
Wallis'
expression becomes , and
points

now coincide at point , as
shown above in Figure 1.3 (b). For
certain, we can say that the value ""

is the proper choice we use today. So in some sense,
Wallis was the first person to place the point in

it's proper location in the upper half-plane. But
we surely would not want to equate the
points " and

." 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

to
vector .

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 -axis. Figure
1.4 (b) illustrates this idea for the vector labeled
as :

it's length is , and
its *angular displacement* from the positive -axis
is .

**(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 , then and .

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 -axis.

Let's call this vector the *standard unit vector*,
, as illustrated in Figure 1.5.

**Figure
1.5** Wessel's view
of the standard unit vector .

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
*

What is the

radians, as it coincides with the positive -axis. Thus, if vectors and have

respectively, and vector , then the

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

displacement of (which is ) differs from the displacement of (which is ) by . This is the same

amount that the

unit vector (which is ). The

which is the same amount that the

standard unit vector .

**(a)** Multiplication
of two
vectors. **
(b)** The square
root of .

**Figure
1.6** Wessel's
multiplication scheme for vectors.

**Wessel's vector **

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
-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 radians,
so by Wessel's rules again the product vector has an

*angular**
displacement* of radians. Thus,
the product vector is aligned along the -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 . Label
the original vector as .

What do you conclude? Obviously,
that , which
must mean that .
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

,

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
solutions to the equation . In
Section
1.5 we show that

these solutions can be expressed for
certain 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 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 by
identifying it with the point
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 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

if both sides of the equation are defined. Applying this identity to complex numbers leads to

.

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 ,
you will find that a solution to the depressed
cubic is
either

or

,

depending on whether you used an older version

and will find that the solution .

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

.

Simplifying the first term in the expression
for we
have

Simplifying the second term in the expression
for we
have

Therefore,

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 which
is not true for complex numbers. If you use a

computer algebra system to compute 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 ,

.

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

**Conjecture. (The Ferro-Tartaglia
Formulae).** If
and
are real or complex constants used to form

the depressed cubic equation

,

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

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

.

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

in the conjecture, and then determine which one is a solution to the
cubic.

**Example
9.** Given the cubic equation

.

Then the corresponding depressed cubic equation is

.

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

in the conjecture, and then determine ones are solutions to the
cubic.

**The solution of quadratic
equations.**

and
can construct the solutions to the general quadratic equation.

**Quadratic
Formula Exploration.**

**The solution of cubic
equations.**

and
can construct the solutions to the general cubic
equation.

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 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
and
are real or complex constants

used to form the depressed cubic equation

,

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

,

for and
,

where , and .

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
for and for and ,

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

.

Use *Mathematica* to calculate the
nine values
with the Ferro-Tartaglia-Cardano formulae,

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

**Example 11.** Given
Cardano's depressed cubic equation

.

Use *Mathematica* to calculate the
nine values
with the Ferro-Tartaglia-Cardano formulae,

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

**The Cube Root
Fallacy**

Does imply
that
?

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

,

and then make the simplification

,

and with a sigh of "anxiety" obtain

.

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
.

The details are also discussed in the
article "*The
Cube Root Fallacy: Does*
*Imply
that*
?",

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 over
the interval .

Graph of , over the interval , as taught in algebra and calculus.

Have you used
or
or
to graph over
the interval
?

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

The
command to draw the graph of is

The
command to draw the graph is
similar

The resulting graph for that
is drawn is is given below.

Graph
of , over
the interval , as
drawn with
or .

Why isn't the graph drawn
when
? Why doesn't it look like the one from algebra and
calculus ?

The answer is easy. All three computer algebra
programs: *,*
and
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
over the interval it
cannot be plotted, and
and

make the graph blank.

The situation is more bizarre if you
use the computer software .

The
commands to graph of are

First,
will print the following error warning message.

Then,
will draw the following mysterious graph for over
the entire interval ,

Graph of , as drawn with .

Can you tell what is happening here with the graph ? Why is the graph positive when ?

The answer is easy.
is actually graphing , where
is the real part

of the complex number .

**Concluding
Remarks.**

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

three major software programs; ,
and
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**

__Fundamental
Theorem of Algebra__

**Download
this Mathematica Notebook**

**The Next Module
is**

**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