the story of ``imaginary'' numbers and why they are not...

The Story of “Imaginary” Numbers . . .. . . and why they are not imaginary

As Told By Girolamo Cardano

Professor of Medicine (Pavia, and Bologna)

Girolamo Cardano (Pavia, Bologna) The Story of “Imaginary” Numbers

The Story of “Imaginary” Numbers

1. Prelude, and a Challenge

2. The Oath and its Aftermath

3. The Dilemma

4. Bombelli’s Breakthrough

5. Geometric Progress: John Wallis

6. Completing the Jigsaw


Important Players

I Luca Pacioli (1445–1509)

I The first to search for the solution to a general cubic equation:x3 + ax2 + bx + c = 0.

I He eventually thinks that such a solution cannot be found.


Important Players

I Luca Pacioli (1445–1509)I The first to search for the solution to a general cubic equation:

x3 + ax2 + bx + c = 0.

I He eventually thinks that such a solution cannot be found.


Important Players

I Luca Pacioli (1445–1509)I The first to search for the solution to a general cubic equation:

x3 + ax2 + bx + c = 0.I He eventually thinks that such a solution cannot be found.


Important Players

I Scipione del Ferro (1465–1526)

I Professor at the University of BolognaI Although unable to solve the general cubic (x3 + ax2 + bx + c = 0), he

makes progress by solving the depressed cubic: x3 + bx + c = 0.

I Solution: x = 3

√− c

2 +√

c2

4 + b3

27 + 3

√− c

2 −√

c2

4 + b3

27 .

I Rather than tell the world del Ferro keeps his discovery a closelyguarded secret . . . Why? Because in those days at any moment one’sposition at an Italian university was subject to a public challenge by anuninvited outsider, who would present a list of challenge problems tobe solved. Thus, if a mathematical scholar made some discovery, he orshe would keep it secret so as to have an ample supply ofcounter-challenges on hand.

I On his deathbed del Ferro finally gives the solution to . . .


Important Players


I Professor at the University of Bologna

I Although unable to solve the general cubic (x3 + ax2 + bx + c = 0), hemakes progress by solving the depressed cubic: x3 + bx + c = 0.

I Solution: x = 3

√− c

2 +√

c2

4 + b3

27 + 3

√− c

2 −√

c2

4 + b3

27 .




Important Players



makes progress by solving the depressed cubic:

x3 + bx + c = 0.

I Solution: x = 3

√− c

2 +√

c2

4 + b3

27 + 3

√− c

2 −√

c2

4 + b3

27 .




Important Players




I Solution: x = 3

√− c

2 +√

c2

4 + b3

27 + 3

√− c

2 −√

c2

4 + b3

27 .




Important Players




I Solution: x = 3

√− c

2 +√

c2

4 + b3

27 + 3

√− c

2 −√

c2

4 + b3

27 .

I Rather than tell the world del Ferro keeps his discovery a closelyguarded secret . . .

Why? Because in those days at any moment one’sposition at an Italian university was subject to a public challenge by anuninvited outsider, who would present a list of challenge problems tobe solved. Thus, if a mathematical scholar made some discovery, he orshe would keep it secret so as to have an ample supply ofcounter-challenges on hand.



Important Players




I Solution: x = 3

√− c

2 +√

c2

4 + b3

27 + 3

√− c

2 −√

c2

4 + b3

27 .

I Rather than tell the world del Ferro keeps his discovery a closelyguarded secret . . . Why?

Because in those days at any moment one’sposition at an Italian university was subject to a public challenge by anuninvited outsider, who would present a list of challenge problems tobe solved. Thus, if a mathematical scholar made some discovery, he orshe would keep it secret so as to have an ample supply ofcounter-challenges on hand.



Important Players




I Solution: x = 3

√− c

2 +√

c2

4 + b3

27 + 3

√− c

2 −√

c2

4 + b3

27 .

I Rather than tell the world del Ferro keeps his discovery a closelyguarded secret . . . Why? Because in those days at any moment one’sposition at an Italian university was subject to a public challenge by anuninvited outsider, who would present a list of challenge problems tobe solved.

Thus, if a mathematical scholar made some discovery, he orshe would keep it secret so as to have an ample supply ofcounter-challenges on hand.



Important Players




I Solution: x = 3

√− c

2 +√

c2

4 + b3

27 + 3

√− c

2 −√

c2

4 + b3

27 .




Important Players

I Antonio Fior (1506–?)

I A mediocre mathematicianI Issues a public challenge (February 1, 1535) to . . .

I Niccolo Fontana (1499–1557)I AKA Tartaglia (“the stammerer”)I Fior’s challenge: 30 problems—all depressed cubics!I Two days before the challenge is to be judged, Tartaglia discovers the

solution to the depressed cubic.I Tartaglia aces Fior’s challenge. Fior flubs Tartaglia’s challenge, and

vanishes in disgrace.I Sometime later Girolamo Cardano pleads with Tartaglia to show him

the solution to the depressed cubic.I Eventually, Targaglia gives the solution to Cardano . . .

. . . but there was a big condition.


Important Players

I Antonio Fior (1506–?)I A mediocre mathematician

I Issues a public challenge (February 1, 1535) to . . .I Niccolo Fontana (1499–1557)

I AKA Tartaglia (“the stammerer”)I Fior’s challenge: 30 problems—all depressed cubics!I Two days before the challenge is to be judged, Tartaglia discovers the






Important Players

I Antonio Fior (1506–?)I A mediocre mathematicianI Issues a public challenge (February 1, 1535) to . . .







Important Players


I Niccolo Fontana (1499–1557)

I AKA Tartaglia (“the stammerer”)I Fior’s challenge: 30 problems—all depressed cubics!I Two days before the challenge is to be judged, Tartaglia discovers the






Important Players


I Niccolo Fontana (1499–1557)I AKA Tartaglia (“the stammerer”)

I Fior’s challenge: 30 problems—all depressed cubics!I Two days before the challenge is to be judged, Tartaglia discovers the






Important Players


I Niccolo Fontana (1499–1557)I AKA Tartaglia (“the stammerer”)I Fior’s challenge: 30 problems—all depressed cubics!

I Two days before the challenge is to be judged, Tartaglia discovers thesolution to the depressed cubic.

I Tartaglia aces Fior’s challenge. Fior flubs Tartaglia’s challenge, andvanishes in disgrace.

I Sometime later Girolamo Cardano pleads with Tartaglia to show himthe solution to the depressed cubic.

I Eventually, Targaglia gives the solution to Cardano . . .



Important Players



solution to the depressed cubic.

I Tartaglia aces Fior’s challenge. Fior flubs Tartaglia’s challenge, andvanishes in disgrace.





Important Players



solution to the depressed cubic.I Tartaglia aces Fior’s challenge.

Fior flubs Tartaglia’s challenge, andvanishes in disgrace.





Important Players




vanishes in disgrace.





Important Players





the solution to the depressed cubic.




Important Players











3. The Dilemma





Enter Girolamo Cardano

I Girolamo Cardano (1501–1576)I The Oath

I swear to you by the Sacred Gospel, and on my faith as a gentleman,not only never to publish your discoveries, if you tell them to me, but Ialso promise and pledge my faith as a true Christian to put them downin cipher so that after my death no one shall be able to understandthem.

I Sometime later, Cardano begins collaborating with Lodovico Ferrari(1522-1565).

I Using their knowledge that there is solution to the depressed cubic, theyare able to get a solution to the general cubic equation . . . Fantastic!





3. The Dilemma





Cardano Faces a Dilemma

I Cardano realizes he cannot publish his solution to the general cubicequation. Why?

Because doing so would require the publication ofthe solution to the depressed cubic equation, and Cardano is boundby his oath never to publish it.

I Cardano and Ferrari eventually travel to the home of del Ferro, who isnow deceased. His estate is being cleared out, and they want toretrieve his mathematical papers lest they be lost to posterity.

I In those papers Cardano and Ferrari find del Ferro’s solution to thedepressed cubic.



I Cardano realizes he cannot publish his solution to the general cubicequation. Why? Because doing so would require the publication ofthe solution to the depressed cubic equation, and Cardano is boundby his oath never to publish it.

I Cardano and Ferrari eventually travel to the home of del Ferro, who isnow deceased. His estate is being cleared out, and they want toretrieve his mathematical papers lest they be lost to posterity.

I In those papers Cardano and Ferrari find del Ferro’s solution to thedepressed cubic.



I Cardano now realizes that there are actually two people who solvedthe depressed cubic equation, and that del Ferro’s discovery pre-datedTartaglia’s work.

I After much thought, Cardano decides that, since he discovered delFerro’s solution, and since del Ferro and Tartaglia did their workindependenly of each other, he is no longer bound by the oath hemade to Tartaglia.Rationale: if he publishes he will be publishing del Ferro’s result(although he will also credit Tartaglia).

I He publishes his work in what has come to be a famous book, ArsMagna, in 1545. Tartaglia is furious.

I Do you think Cardano’s decision was ethical?




I After much thought, Cardano decides that, since he discovered delFerro’s solution, and since del Ferro and Tartaglia did their workindependenly of each other, he is no longer bound by the oath hemade to Tartaglia.

Rationale: if he publishes he will be publishing del Ferro’s result(although he will also credit Tartaglia).







I He publishes his work in what has come to be a famous book, ArsMagna, in 1545.

Tartaglia is furious.I Do you think Cardano’s decision was ethical?





3. The Dilemma





Bombelli Takes a Big Step

I Rafael Bombelli (1526–1572)Publishes L’Algebra in 1569, where he shows how “imaginary”numbers can be used to get real solutions to cubic equations.

I Recall: x3 + bx + c = 0 has a solution of

x =3

√√√√−c2 +

√c2

4 + b3

27 +3

√√√√−c2 −

√c2

4 + b3

27 .

I Example: x3 − 15x − 4 = 0; using the above formula we getx = 3

√2 + 11

√−1 + 3

√2− 11

√−1.

I Bombelli showed that:3

√2 + 11

√−1 = 2 +

√−1, and 3

√2− 11

√−1 = 2−

√−1.

I Therefore x = (2 +√−1) + (2−

√−1) = 4.

This result was a bombshell!





x =3

√√√√−c2 +

√c2

4 + b3

27 +3

√√√√−c2 −

√c2

4 + b3

27 .


√2 + 11

√−1 + 3

√2− 11

√−1.


√2 + 11

√−1 = 2 +

√−1, and 3

√2− 11

√−1 = 2−

√−1.

I Therefore x = (2 +√−1) + (2−

√−1) =

4.This result was a bombshell!





x =3

√√√√−c2 +

√c2

4 + b3

27 +3

√√√√−c2 −

√c2

4 + b3

27 .


√2 + 11

√−1 + 3

√2− 11

√−1.


√2 + 11

√−1 = 2 +

√−1, and 3

√2− 11

√−1 = 2−

√−1.

I Therefore x = (2 +√−1) + (2−

√−1) = 4.

This result was a bombshell!





x =3

√√√√−c2 +

√c2

4 + b3

27 +3

√√√√−c2 −

√c2

4 + b3

27 .


√2 + 11

√−1 + 3

√2− 11

√−1.


√2 + 11

√−1 = 2 +

√−1, and 3

√2− 11

√−1 = 2−

√−1.

I Therefore x = (2 +√−1) + (2−

√−1) = 4.

This result was a bombshell!Girolamo Cardano (Pavia, Bologna) The Story of “Imaginary” Numbers




3. The Dilemma





John Wallis Makes a Contribution

John WallisContributions from A Treatise of Algebra (1685)

I The Number LineI New Twist to a Standard Geometric Problem:

Construct a triangle determined by two sides andan angle not included between those sides.

I The Standard Solution

α

A

B CE F

a b b




I The Number Line

I New Twist to a Standard Geometric Problem:



α

A

B CE F

a b b




I The Number LineI New Twist to a Standard Geometric Problem:



α

A

B CE F

a b b



John WallisContributions from A Treatise of Algebra (1685)Construct a triangle determined by two sides andan angle not included between those sides.

Geometric representations for −√

b2 − c2 and +√

b2 − c2

, (b > c)

α

A

B CE F

a b b

D = 0x-axis

(−√b2 − c2

) (+√b2 − c2

)

c

Question: What if b < c?





b2 − c2 and +√

b2 − c2, (b > c)

α

A

B CE F

a b b

D = 0x-axis

(−√b2 − c2

) (+√b2 − c2

)

c

Question: What if b < c?





b2 − c2 and +√

b2 − c2, (b > c)

α

A

B CE F

a b b

D = 0x-axis

(−√b2 − c2

) (+√b2 − c2

)

c

Question: What if b < c?Girolamo Cardano (Pavia, Bologna) The Story of “Imaginary” Numbers



Construct a triangle determined by two sidesand an angle not included between those sides.


b2 − c2 and +√

b2 − c2, (b < c)

α

A

B C

E F

ab b

D = 0x-axis

(+√b2 − c2

)c



John WallisContributions from A Treatise of Algebra (1685)Construct a triangle determined by two sidesand an angle not included between those sides.


b2 − c2 and +√

b2 − c2, (b < c)

α

A

B C

E F

ab b

D = 0x-axis

(+√b2 − c2

)c





3. The Dilemma





Caspar Wessel Completes the Jigsaw

Caspar Wessel (1797)I Adding vectors

a + b

a

bcopy of b

x

y

x

y

a

α = displacementr =

length

I Wessel’s contribution: showing how to multiply two vectors.

I The length of the product of two vectors should be the product of theirlengths.

I Question: what should the angular displacement of the product of twovectors be?



Caspar Wessel (1797)I Adding vectors

a + b

a

bcopy of b

x

y

x

y

a

α = displacementr =

length

I Wessel’s contribution: showing how to multiply two vectors.I The length of the product of two vectors should be the product of their

lengths.I Question: what should the angular displacement of the product of two

vectors be?



Caspar Wessel (1797)Multiplication of numbers: If c = ab, then c

a = b = b1 , and c

b = a = a1 .

The ratio of the product to any factor is the same as the ratio of the otherfactor to the number one.

Multiplication of vectors:

The (angular) displacement of the product of two vectors should differ fromthe displacement of any given factor by the same amount that thedisplacement of the other factor differs from the displacement of the vectorthat represents the number one.


Caspar Wessel Completes the JigsawCaspar Wessel (1797)


I Consequence 1: The angular displacement of the product is the sumof the individual angular displacements.

I Consequence 2: the point (0, 1) is the square root of −1.

(!)

x

y

aα

β

α+ β

b

ab

x

y

i =√−1

(0, 1)

i2 = −1

(original vector)

(product vector) π2

π


Caspar Wessel Completes the JigsawCaspar Wessel (1797)


I Consequence 1: The angular displacement of the product is the sumof the individual angular displacements.

I Consequence 2: the point (0, 1) is the square root of −1. (!)

x

y

aα

β

α+ β

b

ab

x

y

i =√−1

(0, 1)

i2 = −1

(original vector)

(product vector) π2

π


Is√−1 Really a Number?

“Why do we call something a ‘number’? Well, perhaps because it hasa—direct—relationship with several things that have hitherto been callednumber; and this can be said to give it an indirect relationship to otherthings we call the same name. And we extend our concept of number as inspinning a thread we twist fiber on fiber. And the strength of the threaddoes not reside in the fact that some one fiber runs through its wholelength, but in the overlapping of many fibers.”*

*Ludwig Wittgenstein (Philosophical Investigations, Aphorism 67)


the story of ``imaginary'' numbers and why they are not...

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