An introduction to Complex Numbers - amfidromie.nl · International School Twente: An introduction to Complex Numbers ... while in Baghdad. The title of his book translates as The

An introduction to

Complex Numbers

Rob Kosman Annelieke de Vos 2016

International School Twente: An introduction to Complex Numbers — Lesson 1

2

Introduction

As little children we learned how to count. It was a

proud moment when we could count all the way to 10 .

For a short time we may have felt like we were done counting, and knew all the numbers there were to know. But soon enough more numbers came our way. We counted on to higher and higher numbers, until we realised there was no end to it, and we might as well

stop. These were the natural numbers .

Still in primary school we also learned about the special number 0 , which in some

definitions is included in the set of natural numbers .

But it didn’t stop there. We kept expanding our number system to include more and more numbers. We went on to learn about negative numbers, thus basically constructing the set of integers . And then fractions, constructing the set of rational numbers . And once again it may have felt like surely we must now know all

numbers there are to know.

But then in secondary school we learned about such numbers as 2 , which turned

out to be irrational and have an infinite non-recurring decimal expansion. Other examples of such numbers we encountered were π and e . All such numbers

together form the set of real numbers . To help us work with the real numbers, we learned very early about a possible geometrical representation of all these numbers: the number line.

Each time we expanded our number system, we basically filled in more gaps on the number line. By adding the irrational numbers, we took care of filling in the last of the gaps. Every point on the number line corresponds exactly to one real number. So surely now we must be finished. There are no more numbers to discover. Eh, well, actually …


3

A Puzzle by Gerolamo Cardano

We start this lesson series with a puzzle that’s a few centuries old. Further on in this lesson series we will then revisit this puzzle. The puzzle was introduced by a 16th century Italian mathematician by the name of Gerolamo Cardano (1501-1576). It is recorded in his influential work Ars Magna on algebra. Most scientific texts at that time were still written in Latin. The title Ars Magna translates as The Great Art, a common title for major scientific works at that time. Cardano’s name translates into Latin as Hieronymus Cardanus, a declination of which you can see on the title page of the original text. Our puzzle can be found in chapter 37 of Cardano’s Ars Magna. In the original text, the problem reads as follows in Latin:

Which is rendered in the English translation by T.R. Witmer as:

It is not entirely clear why Cardano writes “ 30 or 40 ”, but in his solution to the

problem he only focuses on the case of the product being 40 . In our modern-day

notation we would restate the puzzle as follows:

Title page from the original text of Ars Magna by Cardano

Find two numbers, and , whose sum is

and whose product is . That is:


4

Exercises

1 a Does this puzzle remind you of a situation in which you solve a problem like this?

b Try to solve Cardano’s puzzle. Spend no more than 5 minutes on it, and draw a preliminary conclusion.

Cubic Equations vs Quadratic Equations

Next lesson we will start by exploring some of the history of solving cubic equations. A general cubic equation (with real coefficients) can be written in the form:

3 2 0ax bx cx d with , , ,a b c d .

Although solving cubic equations is an interesting topic in itself, it will not be our main focus. Instead, we will see that in the context of solving cubic equations, some mysterious things happen that will raise important questions about our number system. Before we start exploring however, it would be a good idea to activate our prior learning on relevant topics. One way to do this is by making a mind map of relevant concepts.

Exercises

2 a Make a mind map of what you know related to

quadratic equations: 2 0ax bx c with , ,a b c .

b Use the following list of concepts to evaluate and possibly enhance your mind map:

a geometric interpretation of a quadratic equation

different methods of solving quadratic equations

different forms of quadratic expressions

polynomial long division

the factor theorem for polynomials

a method of verification for a root of the equation

the role of the discriminant in the quadratic formula


5

3 a Make a mind map of what you know related to

cubic equations: 3 2 0ax bx cx d with , , ,a b c d .

b Use the following list of concepts to evaluate and possibly enhance your mind map:

a geometric interpretation of a cubic equation

the number of (real) solutions to a cubic equation

different methods of solving special cases of cubic equations

polynomial long division

the factor theorem for polynomials

a method of verification for a root of the equation

4 a Compare and contrast the mind maps you made for quadratic and cubic equations. Pay attention to the following aspects:

similarities and connections between both mind maps

differences between both mind maps

the completeness of your knowledge of quadratic equations

the completeness of your knowledge of cubic equations b Compare and contrast the mind maps you made for quadratic and cubic equations with those made by other students. Pay attention to the following aspects:

knowledge gaps that you do not have in common

knowledge gaps that you do have in common

c Express in your own words what the most important difference is between your knowledge of quadratic equations and your knowledge of cubic equations.


6

Solving the Quadratic

The general solution to quadratic equations has been known for a very long time. It was known for example in ancient Babylon. On the Babylonian cuneiform tablet BM13901 from the British Museum, dated 2000-1600 BC, we find a number of examples of quadratic equations. Although the tablet deals only with specific examples, and despite the fact that algebraic problems and their solutions back then were written down in full sentences, instead of our modern-day compact symbolic notation, it is obvious from the steps in the recorded solutions that the Babylonians were already familiar with the method we call ‘completing the square’. This same knowledge we find in other ancient cultures. How to solve quadratic equations was known for example in ancient India, in ancient Greece, and in the Arabic world. During the beginning of the 9th century for example, the Persian scholar al-Khwarizmi wrote an influential book on mathematics while in Baghdad. The title of his book translates as The Compendious Book on Calculation by Completion and Balancing. The original title reads:

الكتاب المختصر في حساب الجبر والمقابلة

Which transcribes as:

Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala

In his book al-Kwarizmi includes a very systematic discussion of solving quadratic equations.

Cuneiform tablet BM13901 from the British Museum

al-Khwarizmi far left: a page from al-Khwarizmi’s book, illustrating the ‘completing the square’ method


7

Exercises

5 At the end of the 12th and the beginning of the 13th century a lot of

mathematics found its way from the Arabic world to medieval Europe through trade routes with northern Africa, including work by al-Khwarizmi.

The influence of this is felt to this very day. The numerals we still use today ( 0,1,2,3,4,5,6,7,8,9 ) are Hindu-Arabic in origin and

were popularised in 13th century Europe by the Italian Leonardo de Pisa, also known as Fibonnaci, in his influential book on arithmetic Liber Abaci.

We also still use words in mathematics that have Arabic origins. The word ‘algorithm’ is derived from the name al-Khwarizmi itself.

From the transcription of the original title of al-Khwarizmi’s book, can you recognize another mathematical word with Arabic origins?

6 Applying the method of ‘completing the square’ to the general quadratic equation:

2 0ax bx c with , ,a b c

leads to the quadratic formula:

2 4

2

b b acx

a

Show that for 1a this can also be written as:

2

2 2

b bx c


8

Solving the Cubic

Despite the fact that over the centuries many people worked on solving cubic equations, and did in fact solve special cases from time to time, little progress was made in solving general cubic equations until the 16th century. Solving the cubic turned out to be a much tougher nut to crack than solving the quadratic. As late as 1492 the Italian mathematician Pacioli made a bold statement in his comprehensive summary of Renaissance mathematics Summa de arithmetica, geometria, proportioni et proportionalita. He declared that the solution of the cubic equation is “as impossible at the present state of science as the quadrature of the circle.” The ‘quadrature of the circle’, or ‘squaring the circle’, being one of the three classical Greek geometrical problems: how to construct a square with the same area as a given circle, using compasses and straightedge only. But Pacioli couldn’t have been more wrong. Whereas the ‘quadrature of the circle’ was eventually proved to be impossible (in 1882), the problem of solving the cubic would be cracked within the next few decades.

Italian mathematician and Franciscan friar Luca Pacioli (c. 1447-1517), surrounded by geometrical instruments and working from The Elements by Euclid.

This painting from 1495 is traditionally attributed to Jacopo de' Barbari.


9

Del Ferro’s Solution

When the general case of a problem is (still) too hard to solve, it usually is a good idea to first try and solve specific subclasses of the problem. This is exactly what the Italian mathematician Scipione del Ferro managed to do around the beginning of the 16th century. With an ingenious method he found a way to solve cubic equations of the form:

3x px q with , , 0p q p q

This is an example of what we call a ‘depressed cubic’, that is a cubic equation without a quadratic term. Note that both parameters p and q are positive.

Exercises

7 Finding the roots of the ‘depressed cubic’


is equivalent to finding the zeros of the function

3( )f x x px q with , , 0p q p q .

a Using differential calculus (not known yet in Del Ferro’s time), find the derivative of the function f .

b What can we say about the sign of ' ( )f x and what does this

mean for the graph of f ?

c Make a sketch of the graph of f . Pay attention to the y -intercept.

d What can we conclude about the number of solutions to Del Ferro’s ‘depressed cubic’? And what can we say about their sign?

8 To appreciate that Del Ferro’s result was not a mere trivial achievement, spend 5 minutes trying to solve his ‘depressed

cubic’ on your own.

If your curiosity is piqued and you are truly stuck on this problem, you can ask for enrichment material that will guide you through a process of rediscovery that will help you understand how Del Ferro managed to find his result.


10

In exercise 7 we found that Del Ferro’s ‘depressed cubic’


always has exactly one real positive solution. Del Ferro found that this solution is provided by the fearsome-looking formula:

2 3 2 3

3 3

2 2 3 2 2 3

q q p q q px

Exercises

9 a Is there ever a risk in Del Ferro’s formula of having to take a

square root of a negative number?

b Is it necessary in Del Ferro’s formula to take a cube root of a negative number? And does this cause a problem?

10 Let’s consider an example that we can solve with Del Ferro’s

formula: 3 6 20x x .

a First find the (real) solution to this equation by inspection and trial and error.

b Find the (real) solution to this equation with Del Ferro’s formula.

c How easy or hard is it to simplify the answer you found with Del Ferro’s formula by hand?

d Use a calculator to evaluate the result from Del Ferro’s formula, and compare this value to the solution you found by inspection.

Del Ferro’s formula works, but the solutions it provides can be very hard to simplify. But more about that later …


11

Keeping the Secret

Unlike what is common in science today, Del Ferro kept his discovery a secret. This has everything to do with the way mathematicians and other scholars had to make a living back then. Del Ferro was a professor at the university of Bologna. But there was no such thing as tenure. One basically had to keep proving worthy of the position, or lose it to someone else. To find employers (with deep pockets), one had to demonstrate one’s superior skills and knowledge.

As a result it was common for mathematicians to challenge each other in what can only be called a ‘maths duels’. Two competitors would set each other a number of problems, and whoever could solve the most, would win. Though they may also have been something of a sport, by beating others in such ‘duels’ one could build reputation and renown, making it easier to find employ. Therefore it was potentially hugely beneficial for someone to keep a new discovery secret, so as to have an edge over competitors.

This doesn’t mean nothing was published, because publishing new results under one’s own name could also bring fame and renown, much like today. But it was far less natural to make discoveries known to the world as soon as possible. Del Ferro passed his secret formula on to a student of his on his deathbed in 1526. This student then went on to challenge Tartaglia, another Italian mathematician who claimed he could solve certain types of cubic equations. Tartaglia accepted the challenge. Knowing Del Ferro’s renown, Tartaglia had a good idea of which types of problems his challenger would set him. So in preparation he worked tirelessly to solve Del Ferro’s ‘depressed cubic’ on his own. Shortly before the ‘duel’ he did in fact (re)discover Del Ferro’s formula independently, and went on to wipe the floor with his challenger.

Cardano, the 16th century Italian mathematician from our initial puzzle, was also working on cubic equations. Having heard of Tartaglia’s fame in this field, he entreated Tartaglia to let him in on the secret. Finally he managed to sway Tartaglia to spill the beans, according to Tartaglia under a solemn oath that he would keep the result secret and not publish it before Tartaglia would have an opportunity to publish it under his own name first. But Tartaglia never got around to doing so, focusing on other things instead.

Italian mathematician Niccolò Fontana Tartaglia

(c. 1500-1557)

Italian mathematician Gerolamo Cardano (1501-1576)


12

Cardano in the meanwhile managed to extend Del Ferro’s and Tartaglia’s result. First he proved the result, since Tartaglia had only given him the formula (in the form of a poem). Then he also went on to solve the other types of ‘depressed cubics’. And maybe even more importantly, he managed to demonstrate that any general cubic equation through a suitable substitution of variables could be rewritten as an equivalent ‘depressed cubic’. Hence he had basically shown how to solve general cubic equations (although his results were not yet amalgamated into a single formula).

Exercises

11 To appreciate that Cardano’s result was not a mere trivial achievement, spend 5 minutes trying to find a suitable substitution of variables that will allow you to write the cubic

3 2 0x bx cx d

as a an equivalent ‘depressed cubic’

3 0w pw q .

If your curiosity is piqued and you are truly stuck on this problem, you can ask for enrichment material that will guide you through a process of rediscovery that will help you understand how Cardano managed to find his result.

Cardano wanted to publish all his results, and went on to do so, despite his possible oath to Tartaglia. Apparently Cardano had heard that Del Ferro had found Tartaglia’s result first, and even went to the trouble of veryfying this from notebooks that after Del Ferro’s passing away had come into the custody of his family members. Cardano published all results in his Ars Magna, the same book in which he presented our initial puzzle. As is only appropriate, in Ars Magna he gave credit to both Del Ferro and Tartaglia for their results. But this did not stop Tartaglia from being very upset anyway. This reaction was understandable. What Tartaglia probably feared, did in fact come to pass. His result nowadays is usually referred to as Cardano’s formula, although it would be more historically correct to call it the Del Ferro – Tartaglia – Cardano formula.

Cardano’s Ars Magna

We will conclude this lesson by having a look at how Cardano in Ars Magna presents the solution to the problem we worked on in exercise 10. This will help us get used to the very different standards of mathematical notation in the 16th century, in preparation for the next lesson.


13

The cubic we tried to solve in exercise 10 was:

3 6 20x x .

We found the (real) solution by investigation to be 2x . But at the same time we

found this same solution from the Del Ferro – Tartaglia – Cardano formula to be:

3 310 108 10 108x

Which can also be written as:

3 3108 10 108 10x

In Cardano’s Ars Magna it looks like this:

Excerpt from Cardano’s Ars Magna, chapter 11.


14

Exercises

12 Verify that our result from exercise 10:

3 310 108 10 108x

is equivalent to the form in which it appears in Cardano’s Ars Magna:

3 3108 10 108 10x

13 a What is Cardano’s notation for the cubic 3 6 20x x ?

b Make a small dictionary from mathematical notation that Cardano uses to our modern-day notation.

Cardano’s notation Our notation

aequalis …

cubus …

6. rebus …

ṕ. …

ḿ. …

…

… 3

14 a Compare and contrast Cardano’s mathematical notation to

our modern-day notation.

b Considering the differences between these different forms of notation, what impact do you think the choice of notation could have on the development of mathematics?

hint:

the Latin word for root is Radix


15

Bombelli to the Rescue

When Cardano presented his solution of the ‘depressed cubic’ 3 6 20x x in his

Ars Magna, it didn’t escape his attention that the solution is actually 2x . In fact he

mentions this in the text as well. But what he did not do, was show how the value

2x could be retrieved from that much more complicated looking solution:

3 310 108 10 108x

This is where the Italian mathematician Bombelli comes into play, who wrote a book called L’Algebra. This book, which was written in Italian rather than in Latin, was intended to be an easy, accessible and practical guide to algebra, which could even be understood by those without a higher education. In this book Bombelli also discussed the results from Cardano’s Ars Magna and explained how to solve cubic equations. But unlike Cardano, he also discussed how a

simple answer like 2x can actually be

‘hiding’ inside such a horrific looking expression as

3 310 108 10 108x .

Here is Bombelli’s way of reasoning. Suppose that the cube root 3 10 108 can

actually be written as a number in the form a b . At this point we don’t know yet

for sure this is possible, but there is no harm in trying, right? In other words, suppose

that we can choose a and b in such a way that:

3

10 108a b .

If so, then necessarily:

3 10 108 a b

Title page of L’Algebra, written by the Italian mathematician

Rafael Bombelli (c. 1526-1572)


16

Exercises

15 Show that the assumption

3

10 108a b

implies that

3 10 108 a b

16 a Using the binomial expansion, show that

3

10 108a b

implies the system of equations

3

2 2

3 10

(3 ) 108

a ab

a b b

b By considering the product

a b a b

also show that Bombelli’s assumption gives us the equation

2 2b a

c Try to solve the combined system of equations

3

2 2

2

3 10

(3 ) 108

2

a ab

a b b

b a

but spend no more than 5 minutes on it.

If you tried to solve the system of equations from exercise 16 by methods of substitution, you probably noticed the system isn’t being very cooperative. Very quickly we are led back to another cubic equation, and it feels like we are going round in circles. But then Bombelli says, wait a minute! If my assumption is correct, then

3 310 108 10 108 2x a b a b a .

But I also know from inspection that the solution is actually 2x . So that leaves only

one possible value for a , and that is 1a . Substituting that back into my system of

equations immediately gives me 3b .


17

In other words, it looks like:

3

3

10 108 1 3

10 108 1 3

Exercises

17 Show algebraically (and not just by calculator) that indeed

3 10 108 1 3 .

Right now you may be crying foul though. We were trying to simplify

3 310 108 10 108x ,

but in the solution process we used the fact that we already knew that 2x . That

reeks of circular reasoning. Bombelli is cheating! This is in fact true. But our gain is

not so much finding out that 2x , but instead in discovering how the value 2 can be

‘hiding’ in something as horrific looking as

3 310 108 10 108 .

Why this is such an important insight, will become clear shortly.

Another Type of ‘Depressed Cubic’

So far we have been focusing on ‘depressed cubics’ of the form


with 3 6 20x x as an example.

But as we pointed out earlier, Cardano systematically discussed all ‘depressed cubics’ in his Ars Magna. We will now consider ‘depressed cubics’ of the form



18

Exercises

18 This exercise is analogous to exercise 7.

Finding the roots of the ‘depressed cubic’


is equivalent to finding the zeros of the function

3( )f x x px q with , , 0p q p q .

a Using differential calculus, find the derivative of the function f .

b What can we say about the sign of ' ( )f x and what does this

mean for the graph of f ?

c Make a sketch of the graph of f . Pay attention to the y -intercept.

d What can we conclude about the number of solutions to this ‘depressed cubic’? And what can we say about their sign?

For this type of ‘depressed cubic’ Cardano presents the following formula for the positive (real) solution:

2 3 2 3

3 3

2 2 3 2 2 3

q q p q q px

To the right is another fragment from Cardano’s Ars Magna, in this case from chapter 12. Cardano discusses an example here that is of the type we are now considering. On the next page is a fragment from Bombelli’s L’Algebra, in which he discusses the very same example.

Excerpt from Cardano’s Ars Magna, chapter 12.


19

Excerpt from Bombelli’s L’Algebra.


20

Exercises

19 Show how Cardano’s formula for ‘depressed cubics’ of type


can be derived from the earlier Del Ferro formula. 20 Using your dictionary from exercise 13 and the excerpt from

Ars Magna on p.18:

a Decipher which cubic equation Cardano discusses here.

b Decipher what the solution is according to Cardano, and verify his solution using his formula.

21 Extend your dictionary from exercise 13 to include the mathematical notation Bombelli uses, by comparing the fragment

from L’Algebra on p.19 with the fragment from Ars Magna on p.18.

Cardano’s Bombelli’s Our

notation notation notation

aequalis … …

cubus … …

6. rebus … …

ṕ. … …

ḿ. … …

… …

… … 3

22 a Find the positive (real) solution to the equation discussed by Cardano and Bombelli in the fragments on p.18 & p.19 by inspection and trial and error.

b Repeat the same process that was proposed by Bombelli as set out on p.15 & p.16 for the solution to the equation discussed by Cardano and Bombelli in the fragments on p.18 & p.19, and thus greatly simplify the answer.


21

c Verify the correctness of your simplified answer algebraically.

d Verify the correctness of your simplified answer by using your dictionary from exercise 21 and by finding the simplified result provided by Bombelli in the second part of the fragment from L’Algebra on p.19.

23 a Compare and contrast the mathematical notations used by Cardano and Bombelli.

b Argue whose notation is closer to our modern-day mathematical notation.

So far, after starting out with an initial puzzle, we have investigated some interesting historical developments in solving cubic equations. We have seen that Cardano systematically described how to solve each type of cubic equation in his Ars Magna. And we have also seen how Bombelli in his L’Algebra showed how the solutions from Cardano’s formulae can be greatly simplified. But we promised on p.4 that solving cubic equations would not be our main focus. And that instead, we would see that in the context of solving cubic equations, some mysterious things happen that would raise important questions about our number system. So where’s the mystery? So far everything seems to be completely ordinary, at least from the point of view of the number system we use. Sure, we’ve seen some numbers written in horrifically complicated ways, but they are still regular real numbers that live somewhere on the number line. That will change in the next lesson however. Now that we have properly set the stage, you will be able to appreciate the mystery in all its glory …


22

Enter the Mystery

Remember that we started off by looking at ‘depressed cubics’ of the form


and that we used Del Ferro’s formula to find the positive (real) solution to this problem? In exercise 9 we concluded that although in Del Ferro’s formula we need to take a cube root of a negative number, which isn’t a problem, we are never at risk of having to take a square root of a negative number, which within the real number system is impossible. But things change for ‘depressed cubics’ of the form

3x px q with , , 0p q p q .

Cardano’s formula to find the positive (real) solution for this type of cubic is:

2 3 2 3

3 3

2 2 3 2 2 3

q q p q q px

And as we can see, now we do run the risk of having to take the square root of a negative number. It doesn’t happen for every cubic of this type, as it didn’t happen in

the example 3 6 40x x we investigated in the previous lesson. But it can happen

for other examples. No doubt Cardano was aware of this. But in his Ars Magna he carefully avoids using the above formula for such cases. Not Bombelli however. In a brave attempt he tried to get to the bottom of what happens in such cases.

Exercises

24 Consider the fragment from Bombelli’s L’Algebra on p.23.

a Using your dictionary from exercise 21, decipher which example of a cubic equation Bombelli discusses here.

b Verify that applying Cardano’s formula to this example results in having to take square roots out of a negative number.

c By inspection and trial and error, find the positive (real) solution to this equation.

d Analogous to what we did in exercises 7 and 18, relate the solutions (or roots) of this equation to the zeros of a suitable function f .


23

Exercises (continued)

24 e Use the factor theorem and polynomial long division to investigate

possible other zeros of the function f .

f Sketch the graph of the function f .

g Draw conclusions about the number and nature of solutions to our original equation, as presented by Bombelli.



24

So the situation Bombelli finds himself in, is this. We have an equation:

3 15 4x x

which is an example of a ‘depressed cubic’ of type:

3x px q with , , 0p q p q .

This means we’d expect Cardano’s formula:

2 3 2 3

3 3

2 2 3 2 2 3

q q p q q px

or its equivalent form:

2 3 2 3

3 3

2 2 3 2 2 3

q q p q q px

to apply for the positive (real) solution of this equation. But when we try to use this last formula, instead we get the impossible looking:

3 32 121 2 121x .

At the same time, by inspection, we already know that 4x is the postive (real)

solution to this equation. We even know that there are two more (real) solutions to

the equation, both of which are negative, 2 3 2 3x x .

Doesn’t this seem to be quite the mystery? So now Bombelli can do one of two things. Either he can take matters at face value and decide to throw in the towel: I know

the equation has 4x as the positive (real) root, but unfortunately, for some reason

Cardano’s formula breaks down in this case. After all, taking the square root

of 121 is simply nonsense.

Or … he can try to make sense of how

3 32 121 2 121

in some mysterious way is actually equal to 4 .

Bombelli decides to face the mystery head on and to go with the latter option.


25

Bombelli’s Wild Thought

We already saw on pages 15 and 16 that Bombelli showed before how:

3 310 108 10 108 1 3 1 3 2

And similarly in exercises 20-22 we saw that Bombelli demonstrated that:

3 320 392 20 392 2 2 2 2 4 .

Bombelli’s ‘wild thought’ now is to ignore for a moment that square roots of negative numbers seem to make no sense, and to try and use the same (or at least a similar) method to show that in some sense:

3 32 121 2 121 4 .

In analogy to the method he used before, he starts by assuming that somehow:

3 2 121 a b with , 0a b b .

It’s still unclear at this point what on earth b is supposed to represent. So even if

it is some type of meaningful new mathematical object, we don’t know yet how it will behave in arithmetic and algebra. But in analogy to identities such as:

2

3 3 3 3 or

2

12 12 12 12

it would seem reasonable to assume that:

2

b b b b with 0b b .

Exercises

25 In analogy to exercise 15, and using the above algebraic rule,

show how the assumption:

3

2 121a b

implies that

3 2 121 a b


26

26 a In analogy to exercise 16, and using the binomial expansion

in combination with our newfound algebraic rule, show that the assumption:

3

2 121a b

implies the system of equations:

3

2 2

3 2

(3 ) ( ) 121

a ab

a b b

b By considering the product:

a b a b

also show that Bombelli’s assumption gives us the equation:

2 5b a

So closely following his method from before, Bombelli finds that the assumptions:

3 2 121 a b and

2

b b b b

lead to the system of equations:

3

2 2

2

3 2

(3 ) ( ) 121

5

a ab

a b b

b a

Just like before (compare exercise 16c), it’s very hard, if not impossible, to solve this system of equations by method of substitution directly. But Bombelli already knows that the following needs to hold:

3 32 121 2 121 2 4a b a b a

Therefore 2a , and substituting this back into the system of equations immediately

yields 1b . So Bombelli’s conclusion now is:

3 32 121 2 121 2 1 2 1 4


27

Exercises

27 a In analogy to exercise 17, show algebraically (using our newfound algebraic rule) that indeed

3 2 121 2 1 .

b As we already pointed out, if these objects b

(with 0b b ) are indeed some meaningful new

mathematical objects, we don’t know yet how they will behave in arithmetic and algebra.

In exercise 27a you used the algebraic rule

2

b b b b with 0b b .

But you also used other ‘new’ algebraic rules, that are inspired by analogous rules for real numbers.

Analyse your working of exercise 27a and specify exactly which other algebraic rules you tacitly assumed to be true.


28

Time to Take a Step Back

Let’s take a step back. What has happened here? Bombelli, by using a method completely analogous to the ones he used in the

ordinary cases, was able to retrieve the value 4 from the expression

3 32 121 2 121

despite the fact that that expression looked nonsensical at first. In fact, he found that:

3 32 121 2 121 2 1 2 1 4 .

But to be able to achieve that, he had to accept the existence of a new type of mathematical object, new ‘numbers’ in fact:

b with 0b b .

These new ‘numbers’ had to obey certain algebraic rules in order for his reasoning to work.

Bombelli introduced a name and notation for these new ‘numbers’. For example:

7 he called “piu di meno 7”, meaning “plus of minus 7”

and

7 he called “meno di meno 7”, meaning “minus of minus 7”



29

Exercises

28 At the bottom of p.28 is the second half of the fragment we already showed on p.23.

a Knowing what names Bombelli gave to these new ‘numbers’, from this fragment verify that Bombelli indeed stated the result to be:

3 32 121 2 121 2 1 2 1 4 .

b What notation does Bombelli use for his new ‘numbers’?

29 At the bottom of this page are two more fragments from Bombelli’s L’Algebra. The fragment on the left shows the rules of multiplication for positive and negative real numbers. With these rules you are already familiar. The fragment on the right shows the new rules of multiplication for Bombelli’s new ‘numbers’.

a Decipher from the left fragment the rules of multiplication you are already familiar with.

b Decipher from the right fragment the rules of multiplication for Bombelli’s new ‘numbers’.

30 A lot of the ‘new’ rules seem to be carried over directly from the familiar rules for real numbers. But there are exceptions.

Consider the identity:

2 3 2 3 6

a Based on this, what might one expect the result to be of:

2 3

b What should its result be though, according to Bombelli’s rules?

Excerpt from Bombelli’s L’Algebra: the rules of multiplication for real positive and real negative numbers

Excerpt from Bombelli’s L’Algebra: the rules of multiplication for Bombelli’s new ‘numbers’


30

Revisiting our Initial Puzzle by Cardano

In exercises 29 and 30 we explored some of the arithmetical and algebraic rules related to Bombelli’s new ‘numbers’. But we already hinted in exercise 27b that there are even more rules we need to reconsider. Further on in this lesson series, when we have introduced the modern-day notation for Bombelli’s new ‘numbers’, we will revisit this topic and list all the arithmetical and algebraic rules we need. But first let’s revisit our initial puzzle by Cardano. In exercise 1 you probably concluded that this puzzle cannot be solved. And you would be right when we decide to limit ourselves to using real numbers only.

Exercises

31 Once more try to solve Cardano’s puzzle, now allowing yourself

to use Bombelli’s new ‘numbers’.

32 With this solution to Cardano’s puzzle, which quadratic expression are you now able to factorise, which before you were unable to?

A Question of Ontology

At the end of the current lesson, it’s time to ask ourselves a very important question.

What has Bombelli discovered here? Are things like 1 really meaningful new

mathematical objects that behave according to a set of rules? Or are they just some meaningless form of trickery that happens to work out in the case of solving certain cubics? And if Bombelli’s new ‘numbers’ are exactly that, that is numbers, then where are they? They are clearly nowhere to be found on the real number line! Is there some kind of ‘physical’ or geometical interpretation for these new ‘numbers’?

Find two numbers, and , whose sum is

and whose product is . That is:


31

In philosophy we call such questions questions of ontology. Ontology refers to the nature of existence of a thing. Do these numbers really ‘exist’? And if so, in what manner?

Exercises

33 Formulate your own opinion (thus far) on the ontological nature of Bombelli’s new ‘numbers’. Are we on the trail of something meaningful and profound, or are we dealing with meaningless trickery?

It would take over another two centuries before a geometical interpretation for Bombelli’s new ‘numbers’ was found. In all that time the ontological status of these new ‘numbers’ was anything but clear. Even many great mathematicians of that era felt very suspicious of these new ‘numbers’. Although Cardano, unlike Bombelli, had avoided using his formula for cubics when it would involve such ‘numbers’, he did use such ‘numbers’ when he presented the solution to this puzzle of his that we opened with. But he clearly felt not very comfortable doing so. He starts his working with the phrase:

“Putting aside all mental tortures involved, …”

And he concludes his solution with the comment:

“So progresses arithmetic subtlety the end of which, as is said,

is as refined as it is useless.”

And even Bombelli felt uneasy with it all, stating:

“The whole matter seems to rest on sophistry rather than on truth.”

Hardly the words of men who feel they are on the trail of something profound instead of some meaningless trickery. In the next lesson we will explore the attitude towards these new ‘numbers’ by some of the greatest mathematicians of the following centuries. We will see that there is actually a strong parallel with the way negative real numbers were initially received. As late as the 16th century, in the time of Cardano and Bombelli, negative numbers were still treated with suspicion by many. In fact, you have already encountered examples of that attitude in the previous lessons, though at the time you may not have noticed.

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An introduction to Complex Numbers - amfidromie.nl · International School Twente: An introduction to Complex Numbers ... while in Baghdad. The title of his book translates as The