The shape of algebra in the mirrors of mathematics: a visual, computer-aided exploration of elementary algebra and beyond
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| Format: | Book |
| Language: | English |
| Published: |
Hackensack, NJ [u.a.]
World Scientific
2012
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 607 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 9814313599 9789814313599 |
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MARC
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| 020 | |a 9789814313599 |9 978-981-4313-59-9 | ||
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| 245 | 1 | 0 | |a The shape of algebra in the mirrors of mathematics |b a visual, computer-aided exploration of elementary algebra and beyond |c Gabriel Katz ; Vladimir Nodel'man |
| 264 | 1 | |a Hackensack, NJ [u.a.] |b World Scientific |c 2012 | |
| 300 | |a XXIV, 607 S. |b Ill., graph. Darst. |e 1 CD-ROM (12 cm) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Algebra |0 (DE-588)4001156-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
| 689 | 0 | 1 | |a Algebra |0 (DE-588)4001156-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Nodelman, Vladimir |e Sonstige |0 (DE-588)1020117885 |4 oth | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-024543578 | ||
Record in the Search Index
| _version_ | 1804148565665120256 |
|---|---|
| adam_text | Contents
Preface
.................................................................................................
vii
1.
Maps, Models, and the Coordinate Plane
...................................... 1
1.1
Rectangles and the Cartesian Plane: Consider a
Rectangular Point
................................................................... 3
1.1.1
The Length-Height Model
............................................ 9
1.1.2
The Perimeter-Area Model
.......................................... 11
1.2
On Parametric Representations
.............................................. 12
1.3
Modeling the Universe of Lines in the Plane
......................... 13
1.3.1
The Slope-Intercept Model
.......................................... 14
1.3.2
The xy-Intercept Model
................................................ 16
1.3.3
The Coefficient Model
................................................. 17
1.4
Parameterizing the Universe of Ellipses
................................ 19
1.5
On Mappings
......................................................................... 20
1.5.1
How VusuMatica Helps to Connect Alternative
Representations of the Modular Space of Rectangles
.. 24
1.5.2
The Geometry of the Connecting Map
......................... 29
1.6
Reflections on the Approach....
.............................................. 31
2.
The Universe of Quadratic Polynomials, the
Viète
Map, and
its Inverse
....................................................................................... 35
2.1
Quadratic Equations and Quadratic Polynomials
.................. 37
2.1.1
A Minor Digression: Polynomials, Equations, Roots,
and Zeros
...................................................................... 39
2.2
The
Viète
Map for Quadratics
............................................... 39
2.2.1
The
Viète
Map and the Mystery of White Region
....... 40
2.2.2
The Discriminant Curve and its Tangential
Investigations
............................................................... 42
2.3
Playing Dice with
Viète
......................................................... 53
2.4
Completing the Square, Substitution Flows, and other
Quadratic Magic
.................................................................... 54
2.5
The Inverse
Viète
Map
.......................................................... 62
2.5.1
Investigating the Geometry of the
Coefficient-to-Root Map
.............................................. 66
2.5.2
Families in the Coefficient Space under the Inverse
Viète
Map
..................................................................... 69
2.6
Adding aNew Dimension....
.................................................. 71
- to
The Complex Numbers and Other Fields
..................................... 77
3.1
From Integers to Rational Numbers...
.................................... 79
3.2
How to Manufacture the Real Numbers from the
Rational Ones....
..................................................................... 81
3.3
From Real to Complex Numbers
........................................... 83
3.3.1
The Need to Extend Real Numbers and the
Incompleteness of the Root Space
............................... 83
3.3.2
Adding Points of the Coordinate Plane
........................ 86
3.3.3
Challenge of Multiplication
......................................... 86
3.4
Back to the Drawing Board
................................................... 88
3.4.1
Replacing Points by Vectors
........................................ 88
3.4.2
Cartesian and Polar Coordinates
.................................. 89
3.4.3
Adding Vectors Geometrically
..................................... 92
3.4.4
Approaching Multiplication of Plane Vectors in Stages
93
3.5
Issues with the Polar Form. Multivalued Representations
..... 99
3.6
Properties of the Complex Numbers
...................................... 101
3.6.1
The Equality of Two Complex Numbers.....
................ 102
3.6.2
Operations with the Cartesian Representation of
Complex Numbers
....................................................... 102
3.6.3
A Short Historical Note
................................................ 105
3.6.4
Division, Conjugation and Absolute Value
.................. 106
3.6.5
The n-th Roots of Unity........
........................................ 110
3.7
A Short Field Trip....
.............................................................. 115
3.8
Conclusions
............................................................................ 125
The Geometry of Complex Linear Polynomial Mappings
............ 129
4.1
Complex Polynomial Maps Considered
................................ 130
4.2
Plane Geometry and Complex Linear Polynomials
.............. 131
4.3
Addition and Multiplication as Transformations
.................. 134
4.4
Composing the Addition and Multiplication Maps...............
136
4.5
Invariants of the Complex Linear Polynomial Maps
............ 139
4.6
On Distances and Isometries
................................................. 140
4.7
Decomposing Complex Multiplication
................................. 143
4.8
Composing Complex Linear Polynomial Maps....................
146
4.9
A Detour About Orientation
...............................,..,....,,,,,,,,, 150
4.10
How to Reconstruct an Isometry of the Plane from its
Action on Points?
...........................,.,.,.........,.,.....,.,,,.,.,...,,,.. 152
4.11
Reflections
....................................................................,..,.,,. 155
4.12
The Structure of Complex Linear Polynomial
Transformations
.................................................................... 157
4.13
The Group, a Fundamental Algebraic Structure
................... 158
4.13.1
Old Familiar Examples of Groups
............................. 159
4.13.2
Subgroups of the Group
L
......................................... 160
4.14
Fixed Points of Linear Polynomial Maps
.............................. 164
Complex Polynomial Maps, Closed Plane Curves, and a Few
Topological Exhibits
..................................................................... 173
5.1
Polynomial Equations and Maps
........................................... 174
5.1.1
From Polynomial Maps to Plane Curves
................... 176
5.1.2
Loops within Loops
................................................... 178
5.2
The World of Plane Curves
................................................... 179
5.2.1
The Rope Model.......
................................................. 182
5.2.2
Self-intersections
....................................................... 183
5.2.3
Simplifying Intersections....
....................................... 186
5.2.4
Identifying Intersection Points on a Plane Curve.
,,,,. 18/
5.2.5
Moving the Rope and Deforming the Curve
,,,,.,,,,.... 190
5.2.6
The Viewpoint of a Nail
............................................
J.9V,
5.2.7
The Cutting Number.,,,....,.,.,,.,.,.,..,,.,...,.,.,.,.,.,,,,,.,.,.,
193
5.2.8
The Algebraic Cutting Number
.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 196
5.3
The Train Model
—
Parameterizing Plane Curves
............... 198
5.3.1
Loops of Traveling Light
........................................... 199
5.3.2
Motion and Angular Accumulation
........................... 201
5.3.3
Index of a Curve with Respect to a Point
—
the
Winding Number
....................................................... 206
5.3.4
The Cutting Number and the Index
........................... 208
5.3.5
Index
Invariance
under Deformations
....................... 214
5.4
The Fundamental Group, the Index Invariant, and a Crush
Course in Topology
............................................................... 219
5.4.1
A Light Speed Flight through the Language
of Topology
............................................................... 219
5.4.2
The Fundamental Group
............................................ 225
Geometry of Complex Polynomial Maps of Degree
Two and Three
.............................................................................. 255
6.1
Zeros and Preimages of Complex Polynomial Maps
............256
6.2
Geometrical Tools for Locating Preimages of Points under
Complex Polynomial Maps
...................................................257
6.3
The Quadratic Map
...............................................................258
6.4
Root and Coefficient Representations
...................................261
6.5
Lassoing Roots
...................................................................... 262
6.5.1
Reflections on Exercise
6.3....................................... 264
6.6
The Quadratic Polynomial with Roots of Multiplicity
2......265
6.6.1
Reflections on Exercise
6.4....................................... 266
6.7
The Local Geometry of Complex Quadratic Maps
............... 266
6.7.1
Reflections on Exercise
6.5....................................... 269
6.7.2
More Reflections
....................................................... 274
6.8
The Monic Cubic Polynomial Equation
................................ 276
Modular Spaces of Cubic and Quartic Polynomials
..................... 285
7.1
Cubic Polynomials and the
Viète
Map
.................................. 286
7.2
Cubic Polynomials and their Discriminants
.......................... 296
7.2.1
Playing Dice with the Real Cubic
Viete
...................... 300
7.3
The Discriminant Surface: Stratified and Ruled
................... 302
7.4
The Reduction Flow and the Depressed
Cubic
Polynomials
319
7.5
The Whitney Projection: Adding Dimension to the
Coefficient Space
.................................................................. 323
7.6
A High Altitude Flight over the Quartic Planet: Surveying
the Discriminant Geometry of Quartic Polynomials.............
329
I. Cubic and Quartic Equations: Solutions in Radicals and a
Field Theory
Trip
.......................................................................... 337
8.1
Radicals as Multi-valued Functions
...................................... 338
8.2
Cubic Formula: the Renaissance Art of Solving Cubic
Equations
............................................................................... 338
8.3
Ferrari s Quartic Formula
..................................................... 347
8.4
Algebra Fields Forever
......................................................... 350
8.5
What Kind of Fields Do the Cubic and Quartic
Polynomials Grow?
............................................................... 364
8.6
More History of Cubic Equations than You Want
to Know
................................................................................. 368
8.6.1
The Case of the Disappearing Proof
.......................... 371
8.6.2
Viète
Comes Back
..................................................... 376
9.
Solutions of Cubic Equations in Radicals: their Geometry
and Symmetry
............................................................................... 379
9.1
Radicals,
Multi-
valued Functions, and Riemann Surfaces
... 380
9.2
Visualizing the Cubic Formula: the
Cardano
Map and the
Ѕз
-symmetry.
......................................................................... 395
10.
The Cubic and Quartic Equations and the Plane Curves...............
447
10.1
How Algebraic Geometry Does Not Help to Solve Cubic
Equations, but Gives Hope
448
10.2
How Algebraic Geometry Does Help to Solve Quartic
Equations....
........................................................................... 459
11.
Complex Polynomial Maps of High Degrees and the
Fundamental Theorem of Algebra
................................................ 46/
11.1
The Two Faces of Polynomials.............
............................... 468
xxiv
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опаре
oj
/itgeuru
11.2
Lassoing the Preimages of Points under Complex
Polynomial Maps
.................................................................. 477
11.3
The Taylor Formula for Polynomials
....................................484
11.3.1
Planets, Generalized Cycloids, and Complex
Polynomials
............................................................... 485
11.3.2
Concluding the Proof of Theorem A
.........................490
11.4
The Fundamental Theorem of Algebra and its Implications.
495
11.5
Proof of the Fundamental Theorem of Algebra
.................... 503
11.6
The Lady with a Dog is enjoying her new algebraic life...
507
11.7
Estimating Polynomials and Measuring their Tails
.............. 510
11.7.1
Applying Estimates to a Specific Polynomial
........... 513
11.8
TFA: Summary and Comments
............................................ 514
11.9
The Fancy Fundamental Theorem and the Complex
Viète
Map
................................................................................... 515
11.9.1
Deep Reflections on the
Viète
Formulas and the
Fancy FTA
................................................................. 518
11.10
Braids and Roots
.................................................................525
11.11
The Galois Theory and the Abel-Ruffini Theorem
............. 530
12.
The Polynomial Kalidoscope with Holomorphic Mirrors
............. 547
12.1
The Derivative
...................................................................... 548
12.2
Roots of a polynomial and its derivative
..............................552
12.3
More about the Taylor Formula for Polynomials
................. 559
12.4
Geometry of complex polynomial maps in the vicinity of
their singular points....
........................................................... 563
12.5
Fixed and Periodic Points of Polynomial Maps
.................... 570
12.6
Polynomial Maps, Vector Fields and Fixed Points...............
575
12.7
The Complex Exponential Map
............................................ 583
12.8
Commuting Polynomial Maps
.............................................. 588
12.9
Line Integrals and Cauchy Formulas: an Analytical
Proof of Theorem A
.............................................................. 591
Reference
............................................................................................. 603
|
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| id | DE-604.BV039694906 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:09:10Z |
| institution | BVB |
| isbn | 9814313599 9789814313599 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024543578 |
| oclc_num | 767775035 |
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| owner | DE-739 DE-824 |
| owner_facet | DE-739 DE-824 |
| physical | XXIV, 607 S. Ill., graph. Darst. 1 CD-ROM (12 cm) |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Katz, Gabriel 1948- Verfasser (DE-588)1020116749 aut The shape of algebra in the mirrors of mathematics a visual, computer-aided exploration of elementary algebra and beyond Gabriel Katz ; Vladimir Nodel'man Hackensack, NJ [u.a.] World Scientific 2012 XXIV, 607 S. Ill., graph. Darst. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 s Algebra (DE-588)4001156-2 s DE-604 Nodelman, Vladimir Sonstige (DE-588)1020117885 oth Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024543578&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Katz, Gabriel 1948- The shape of algebra in the mirrors of mathematics a visual, computer-aided exploration of elementary algebra and beyond Numerisches Verfahren (DE-588)4128130-5 gnd Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4001156-2 |
| title | The shape of algebra in the mirrors of mathematics a visual, computer-aided exploration of elementary algebra and beyond |
| title_auth | The shape of algebra in the mirrors of mathematics a visual, computer-aided exploration of elementary algebra and beyond |
| title_exact_search | The shape of algebra in the mirrors of mathematics a visual, computer-aided exploration of elementary algebra and beyond |
| title_full | The shape of algebra in the mirrors of mathematics a visual, computer-aided exploration of elementary algebra and beyond Gabriel Katz ; Vladimir Nodel'man |
| title_fullStr | The shape of algebra in the mirrors of mathematics a visual, computer-aided exploration of elementary algebra and beyond Gabriel Katz ; Vladimir Nodel'man |
| title_full_unstemmed | The shape of algebra in the mirrors of mathematics a visual, computer-aided exploration of elementary algebra and beyond Gabriel Katz ; Vladimir Nodel'man |
| title_short | The shape of algebra in the mirrors of mathematics |
| title_sort | the shape of algebra in the mirrors of mathematics a visual computer aided exploration of elementary algebra and beyond |
| title_sub | a visual, computer-aided exploration of elementary algebra and beyond |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Algebra (DE-588)4001156-2 gnd |
| topic_facet | Numerisches Verfahren Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024543578&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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