why was caspar wessel s geometrical representation of the complex numbers ignored at his time l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Why was Caspar Wessel’s geometrical representation of the complex numbers ignored at his time? PowerPoint Presentation
Download Presentation
Why was Caspar Wessel’s geometrical representation of the complex numbers ignored at his time?

Loading in 2 Seconds...

play fullscreen
1 / 32

Why was Caspar Wessel’s geometrical representation of the complex numbers ignored at his time? - PowerPoint PPT Presentation


  • 473 Views
  • Uploaded on

Why was Caspar Wessel’s geometrical representation of the complex numbers ignored at his time?. Qu’est-ce que la géométrie? Luminy 17 April 2007 Kirsti Andersen The Steno Institute Aarhus University. Programme 1. Wessel’s work 2. Other similar works 3.Gauss’s approach to complex numbers

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Why was Caspar Wessel’s geometrical representation of the complex numbers ignored at his time?' - Anita


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
why was caspar wessel s geometrical representation of the complex numbers ignored at his time

Why was Caspar Wessel’s geometrical representation of the complex numbers ignored at his time?

Qu’est-ce que la géométrie?

Luminy 17 April 2007

Kirsti Andersen

The Steno Institute

Aarhus University

slide2

Programme

1. Wessel’s work

2. Other similar works

3.Gauss’s approach to complex numbers

4. Cauchy and Hamilton avoiding geometrical interpretations

5. Concluding remarks

slide3

References

Jeremy Gray, “Exkurs: Komplexe Zahlen” in Geschichte der Algebra, ed. Erhard Scholz, 293–299, 1990.

Kirsti Andersen, “Wessel’s Work on Complex Numbers and its Place in History” in Caspar Wessel, On the Analytical Representation of Direction, ed. Bodil Branner and Jesper Lützen, 1999.

slide4

Short biography of Caspar Wessel

Born in Vestby, Norway, as son of a minister 1745

Started at the grammar school in Christiania, now Oslo, in 1757

Examen philosophicum at Copenhagen University 1764

Assistant to his brother who was a geographical surveyor

From 1768 onwards cartographer, geographical surveyor, trigonometrical surveyor

Surveying superintendent, 1798

slide5

Short biography of Caspar Wessel

1778 Exam in law – he never used it

1787 Calculations with expressions of the form

1797 Presentation in the Royal Danish Academy of Sciences and Letters of On the Analytical Representation of Direction. An Attempt Applied Chiefly to Solving Plane and Spherical Polygons

1799 Publication of Wessel’s paper in the Transactions of the Academy

1818 Wessel died without having become a member of the Academy

slide7

Short biography of Caspar Wessel

He calculated the sides in a lot of plane and spherical triangles

He wondered whether he could find a shortcut

slide8

On the Analytical Representation of Direction

Wessel’s aim: an algebraic technique for dealing with directed line segments

In his paper he first looked at a plane, in which he defined addition and multiplication

Addition: the parallelogram rule

slide9

On the Analytical Representation of Direction

His definition of multiplication – could be inspired by Euclid, defintion VII.15

A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced

Or in other words the product is formed by the one factor as the other is formed by the unit

slide10

On the Analytical Representation of Direction

Wessel introduced a unit oe, and required

the product of two straight lines should in every respect be formed from the one factor, in the same way as the other factor is formed from the ... unit

Advanced for its time? Continuation of Descartes?

< eoc = <eoa + <eob

|oc |: |ob |= |oa |: |oe |

or

|oc | = |oa | ⋅ |ob |

slide11

On the Analytical Representation of Direction

e

Next, Wessel introduced another unit and could then express any directed line segment as

The multiplication rule implies that

a

+

b

e

r

(cos

u

+

sin

u

e

)

=

e

×

e

=

-

1

slide12

On the Analytical Representation of Direction

For

The addition formulae

cos(u+v) = cosucosv + sinusinv

sin(u+v) = cosusinv + sinucosv:

a

+

b

e

=

r

(cos

u

+

sin

u

e

)

c

+

d

e

=

r

'

(cos

v

+

sin

v

e

)

(

a

+

b

e

)(

c

+

d

e

)

=

rr

'

(cos(

u

+

v

)

+

sin(

u

+

v

)

e

)

(

a

+

b

e

)(

c

+

d

e

)

=

(

ac

-

bd

)

+

(

ad

+

bc

)

e

slide13

On the Analytical Representation of Direction

Wessel: no need to learn new rules for calculating

He thought he was the first to calculate with directed line segments

Proud and still modest

As he also worked with spherical triangles he would like to work in three dimensions

He was not been able to do this algebraically, but he did not give up

slide14

On the Analytical Representation of Direction

h

A second imaginary unit ,

When is rotated the angle v around the - axis, the result is

and rotating the angle u around the -axis gives a similar expression

In this way he avoided

h

y

h

+

x

¢+

z

'

e

=

y

h

+

(cos

v

+

e

sin

v

)(

x

+

z

e

)

e

h

slide15

On the Analytical Representation of Direction

First a turn of the sphere the outer angle A around the η-axis

slide16

On the Analytical Representation of Direction

On so on six times, until back in starting position

slide17

On the Analytical Representation of Direction

Both in the case of plane polygons and spherical polygons Wessel deduced a neat universal formula

However, solving them were in general not easier than applying the usual formulae

slide18

On the Analytical Representation of Direction

Summary on Wessel’s work

He searched for an algebraic technique for calculating with directed line segment

As a byproduct, he achieved a geometrical interpretation of the complex numbers. He did not mention this explicitly

However, a cryptic remark about that the possible sometimes can be reached by “impossible operations”

slide19

Reaction to Wessel’s work

Nobody took notice of Wessel’s paper

Why?

Among the main stream mathematicians no interest for the geometrical interpretation of complex numbers in the late 18th and the beginning of the 19th century!

slide20

Signs of no interest

○ If the geometrical interpretation of complex numbers had been considered a big issue, Wessel’s result would have been noticed

○ After Wessel, several other interpretations were published, they were not noticed either

○ Gauss had the solution, but did not find it worth while to publish it

○ Cauchy and Hamilton explicitly were against a geometrical interpretation

slide21

Other geometrical interpretations

Abbé Buée 1806

Argand 1806, 1813

Jacques Frédéric Français 1813

Gergonne 1813

François Joseph Servois’s reaction in 1814

no need for a masque géométrique

directed line segments with length a and direction angle α descibed by a function φ (a,α) with certain obvious properties

|

has these, but there could be more functions

j

j

a

-

1

(

a

,

a

)

=

ae

slide22

Other geometrical interpretations

Benjamin Gompertz 1818

John Warren 1828

C.V. Mourey 1828

slide23

Gauss

Gauss claimed in 1831 that already in 1799 when he published his first proof of the fundamental theorem of algebra he had an understanding of the complex plane

In 1805 he made a drawing in a notebook indicating he worked with the complex plane

A letter to Bessel from 1811 (on “Cauchy integral theorem”) shows a clear understanding of the complex plane

However, he only let the world know about his thoughts about complex numbers in a paper on complex integers published in 1831

slide24

Cauchy

Cauchy Cours d’analyse (1821)

an imaginary equation is only a symbolic representation of two equations between real quantities

26 years later he was still of the same opinion. He then wanted to avoid

the torture of finding out what is represented by the symbol , for which the German geometers substitute the letter i

slide25

Cauchy

Instead he chose – an interesting for the time – introduction based on equivalence classes of polynomials

[mod ]

when the the two first polynomials have the same remainder after division by the polynomial

He then introduced i, and rewrote the above equation as

j

(

x

)

º

c

(

x

)

w

(

x

)

j

(

i

)

=

c

(

i

)

slide26

Cauchy

Setting

he had found an explanation why

2

w

(

x

)

=

x

+

1

(

a

+

bi

)(

c

+

di

)

=

ac

-

bd

+

(

ad

+

bc

)

i

slide27

Cauchy

By 1847 Cauchy had made a large part of his important contributions to complex function theory – without acknowledging the complex plane

Later the same year, however, he accepted the geometrical interpretation

slide28

Hamilton

Similar to Cauchy’s couples of real numbers

Hamilton introduced complex numbers as a pair of real numbers in 1837– unaware at the time of Cauchy’s approach

He wished to give square roots of negatives a meaning

without introducing considerations so expressly geometrical, as those which involve the conception of an angle

slide29

Hamilton

His approach went straightforwardly until he had to determine the γs in

Introducing a requirement corresponding to that his multiplication should not open for zero divisors he found the necessary and sufficient condition that

and then concluded that this could be obtained by setting and

(

0

,

1

)

×

(

0

,

1

)

=

(

g

,

g

)

1

2

1

2

g

+

g

<

0

1

4

g

=

0

g

=

-

1

2

1

slide30

Hamilton

In other words Hamilton preferred an inconclusive algebraic argument to a geometrical treatment

slide31

Concluding remarks

When the mathematicians in the seventeenth century struggled with coming to terms with complex numbers a geometrical interpretation would have been welcome

It might for instance have helped Leibniz in his confusion about

By the end of the eighteenth century there was the idea that analytical/algebraic problems should be solved by

analytical/algebraic methods. Hence no interest for Wessel’s and others’ interpretations of complex numbers

slide32

Concluding remarks

A geometrical interpretation could at most be considered an illustration, not a foundation

Warren in 1829 about the reaction to his book from 1828

... it is improper to introduce geometric considerations into questions purely algebraic; and that the geometric representation, if any exists, can only be analogical, and not a true algebraic representation of the roots