Complex numbers from A to ... Z:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2006
|
| Ausgabe: | Greatly expanded and substantially enhanced version of the Romanian ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XI, 321 S. graph. Darst. |
| ISBN: | 0817643265 9780817643263 |
Internformat
MARC
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| 100 | 1 | |a Andreescu, Titu |d 1956- |e Verfasser |0 (DE-588)12217805X |4 aut | |
| 240 | 1 | 0 | |a Numere complexe de la A la ... Z |
| 245 | 1 | 0 | |a Complex numbers from A to ... Z |c Titu Andreescu ; Dorin Andrica |
| 250 | |a Greatly expanded and substantially enhanced version of the Romanian ed. | ||
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2006 | |
| 300 | |a XI, 321 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
| _version_ | 1804135089257316352 |
|---|---|
| adam_text | Contents
Preface
ix
Notation xiii
1
Complex
Numbers in Algebraic Form
1
1.1
Algebraic Representation of Complex Numbers
............ 1
1.1.1
Definition of complex numbers
................. 1
1.1.2
Properties concerning addition
................. 2
1.1.3
Properties concerning multiplication
.............. 3
1.1.4
Complex numbers in algebraic form
.............. 5
1.1.5
Powers of the number
і
..................... 7
1.1.6
Conjugate of a complex number
................ 8
1.1.7
Modulus of a complex number
................. 9
1.1.8
Solving quadratic equations
................... 15
1.1.9
Problems
............................ 18
1.2
Geometric Interpretation of the Algebraic Operations
......... 21
1.2.1
Geometric interpretation of a complex number
......... 21
1.2.2
Geometric interpretation of the modulus
............ 23
1.2.3
Geometric interpretation of the algebraic operations
...... 24
1.2.4
Problems
............................ 27
vi
Contents
2
Complex
Numbers in Trigonometric Form
29
2.1
Polar Representation of Complex Numbers
.............. 29
2.1.1
Polar coordinates in the plane
.................. 29
2.1.2
Polar representation of a complex number
........... 31
2.1.3
Operations with complex numbers in polar representation
. . . 36
2.1.4
Geometric interpretation of multiplication
........... 39
2.1.5
Problems
............................ 39
2.2
The nih Roots of Unity
......................... 41
2.2.1
Defining the nth roots of a complex number
.......... 41
2.2.2
The nih roots of unity
...................... 43
2.2.3
Binomial equations
....................... 51
2.2.4
Problems
............................ 52
3
Complex Numbers and Geometry
53
3.1
Some Simple Geometric Notions and Properties
............ 53
3.1.1
The distance between two points
................ 53
3.1.2
Segments, rays and lines
.................... 54
3.1.3
Dividing a segment into a given ratio
.............. 57
3.1.4
Measure of an angle
....................... 58
3.1.5
Angle between two lines
.................... 61
3.1.6
Rotation of a point
....................... 61
3.2
Conditions for Collinearity, Orthogonality and Concyclicity
...... 65
3.3
Similar Triangles
............................ 68
3.4
Equilateral Triangles
.......................... 70
3.5
Some Analytic Geometry in the Complex Plane
............ 76
3.5.1
Equation of a line
........................ 76
3.5.2
Equation of a line determined by two points
.......... 78
3.5.3
The area of a triangle
...................... 79
3.5.4
Equation of a line determined by a point and a direction
. ... 82
3.5.5
The foot of a perpendicular from a point to a line
....... 83
3.5.6
Distance from a point to a line
................. 83
3.6
The Circle
................................ 84
3.6.1
Equation of a circle
....................... 84
3.6.2
The power of a point with respect to a circle
.......... 86
3.6.3
Ansile
between two circles
................... 86
Contents
vii
4
More on Complex Numbers and Geometry
89
4.1
The Real Product of Two Complex Numbers
.............. 89
4.2
The Complex Product of Two Complex Numbers
........... 96
4.3
The Area of a Convex Polygon
..................... 100
4.4
Intersecting Cevians and Some Important Points in a Triangle
..... 103
4.5
The Nine-Point Circle of
Euler
..................... 106
4.6
Some Important Distances in a Triangle
................ 110
4.6.1
Fundamental invariants of a triangle
.............. 110
4.6.2
The distance OI
......................... 112
4.6.3
The distance ON
........................ 113
4.6.4
The distance OH
........................ 114
4.7
Distance between Two Points in the Plane of a Triangle
........ 115
4.7.1
Barycentric coordinates
..................... 115
4.7.2
Distance between two points in barycentric coordinates
.... 117
4.8
The Area of a Triangle in Barycentric Coordinates
........... 119
4.9
Orthopolar Triangles
.......................... 125
4.9.1
The Simson-Wallance line and the pedal triangle
....... 125
4.9.2
Necessary and sufficient conditions for orthopolarity
..... 132
4.10
Area of the
Antipedal
Triangle
..................... 136
4.11
Lagrange s Theorem and Applications
................. 140
4.12
Euler s Center of an Inscribed Polygon
................. 148
4.13
Some Geometric Transformations of the Complex Plane
....... 151
4.13.1
Translation
........................... 151
4.13.2
Reflection in the real axis
................... 152
4.13.3
Reflection in a point
...................... 152
4.13.4
Rotation
............................ 153
4.13.5
Isometric transformation of the complex plane
........ 153
4.13.6
Morley s theorem
....................... 155
4.13.7
Homothecy
........................... 158
4.13.8
Problems
............................ 160
5
Olympiad-Caliber Problems
161
5.1
Problems Involving Moduli and Conjugates
.............. 161
5.2
Algebraic Equations and Polynomials
................. 177
5.3
From Algebraic Identities to Geometric Properties
........... 181
5.4
Solving Geometric Problems
...................... 190
5.5
Solving Trigonometric Problems
.................... 214
5.6
More on the ntb Roots of Unity
..................... 220
viii Contents
5.7 Problems
Involving
Polygons...................... 229
5.8
Complex Numbers and Combinatorics
................. 237
5.9
Miscellaneous Problems
........................ 246
6
Answers, Hints and Solutions to Proposed Problems
253
6.1
Answers, Hints and Solutions to Routine Problems
.......... 253
6.1.1
Complex numbers in algebraic representation (pp.
18-21) . . . 253
6.1.2
Geometric interpretation of the algebraic operations (p.
27) . . 258
6.1.3
Polar representation of complex numbers (pp.
39-41)..... 258
6.1.4
The nlb roots of unity (p.
52).................. 260
6.1.5
Some geometric transformations of the complex plane (p.
160) 261
6.2
Solutions to the Olympiad-Caliber Problems
.............. 262
6.2.1
Problems involving moduli and conjugates (pp.
175-176) . . . 262
6.2.2
Algebraic equations and polynomials (p.
181)......... 269
6.2.3
From algebraic identities to geometric properties (p.
190) . . . 272
6.2.4
Solving geometric problems (pp.
211-213)........... 274
6.2.5
Solving trigonometric problems (p.
220)............ 287
6.2.6
More on the
и 1
roots of unity (pp.
228-229).......... 289
6.2.7
Problems involving polygons (p.
237) ............. 292
6.2.8
Complex numbers and combinatorics (p.
245)......... 298
6.2.9
Miscellaneous problems (p.
252)................ 302
Glossary
307
References
313
Index of Authors
317
Subject Index
319
Complex
Numbers
2,
It is impossible to imagine modern mathematics without complex
numbers. Complex Numbers from A to...
Z
introduces the reader
to this fascinating subject which, from the time of
Euler,
has
become one of the most utilized in mathematics.
The exposition concentrates on key concepts and then elemen¬
tary results concerning these numbers. The reader learns how
complex numbers can be used to solve algebraic equations and
to understand the geometric interpretation of complex numbers
and the operations involving them.
The theoretical parts of the book are augmented with rich exer¬
cises and problems at various levels of difficulty. A special fea¬
ture of the book is the last chapter, a selection of outstanding
Olympiad and other important mathematical contest problems
solved by employing the methods already presented.
The book reflects the unique experience of the authors. It distills
a vast mathematical literature, most of which is unknown to the
western public, and captures the essence of an abundant prob¬
lem culture. The target audience includes undergraduates, high
school students and their teachers, mathematical contestants
(such as those training for Olympiads or the W. L. Putnam
о
Mathematical Competition) and their coaches, as well as anyone
interested in essential mathematics.
|
| adam_txt |
Contents
Preface
ix
Notation xiii
1
Complex
Numbers in Algebraic Form
1
1.1
Algebraic Representation of Complex Numbers
. 1
1.1.1
Definition of complex numbers
. 1
1.1.2
Properties concerning addition
. 2
1.1.3
Properties concerning multiplication
. 3
1.1.4
Complex numbers in algebraic form
. 5
1.1.5
Powers of the number
і
. 7
1.1.6
Conjugate of a complex number
. 8
1.1.7
Modulus of a complex number
. 9
1.1.8
Solving quadratic equations
. 15
1.1.9
Problems
. 18
1.2
Geometric Interpretation of the Algebraic Operations
. 21
1.2.1
Geometric interpretation of a complex number
. 21
1.2.2
Geometric interpretation of the modulus
. 23
1.2.3
Geometric interpretation of the algebraic operations
. 24
1.2.4
Problems
. 27
vi
Contents
2
Complex
Numbers in Trigonometric Form
29
2.1
Polar Representation of Complex Numbers
. 29
2.1.1
Polar coordinates in the plane
. 29
2.1.2
Polar representation of a complex number
. 31
2.1.3
Operations with complex numbers in polar representation
. . . 36
2.1.4
Geometric interpretation of multiplication
. 39
2.1.5
Problems
. 39
2.2
The nih Roots of Unity
. 41
2.2.1
Defining the nth roots of a complex number
. 41
2.2.2
The nih roots of unity
. 43
2.2.3
Binomial equations
. 51
2.2.4
Problems
. 52
3
Complex Numbers and Geometry
53
3.1
Some Simple Geometric Notions and Properties
. 53
3.1.1
The distance between two points
. 53
3.1.2
Segments, rays and lines
. 54
3.1.3
Dividing a segment into a given ratio
. 57
3.1.4
Measure of an angle
. 58
3.1.5
Angle between two lines
. 61
3.1.6
Rotation of a point
. 61
3.2
Conditions for Collinearity, Orthogonality and Concyclicity
. 65
3.3
Similar Triangles
. 68
3.4
Equilateral Triangles
. 70
3.5
Some Analytic Geometry in the Complex Plane
. 76
3.5.1
Equation of a line
. 76
3.5.2
Equation of a line determined by two points
. 78
3.5.3
The area of a triangle
. 79
3.5.4
Equation of a line determined by a point and a direction
. . 82
3.5.5
The foot of a perpendicular from a point to a line
. 83
3.5.6
Distance from a point to a line
. 83
3.6
The Circle
. 84
3.6.1
Equation of a circle
. 84
3.6.2
The power of a point with respect to a circle
. 86
3.6.3
Ansile
between two circles
. 86
Contents
vii
4
More on Complex Numbers and Geometry
89
4.1
The Real Product of Two Complex Numbers
. 89
4.2
The Complex Product of Two Complex Numbers
. 96
4.3
The Area of a Convex Polygon
. 100
4.4
Intersecting Cevians and Some Important Points in a Triangle
. 103
4.5
The Nine-Point Circle of
Euler
. 106
4.6
Some Important Distances in a Triangle
. 110
4.6.1
Fundamental invariants of a triangle
. 110
4.6.2
The distance OI
. 112
4.6.3
The distance ON
. 113
4.6.4
The distance OH
. 114
4.7
Distance between Two Points in the Plane of a Triangle
. 115
4.7.1
Barycentric coordinates
. 115
4.7.2
Distance between two points in barycentric coordinates
. 117
4.8
The Area of a Triangle in Barycentric Coordinates
. 119
4.9
Orthopolar Triangles
. 125
4.9.1
The Simson-Wallance line and the pedal triangle
. 125
4.9.2
Necessary and sufficient conditions for orthopolarity
. 132
4.10
Area of the
Antipedal
Triangle
. 136
4.11
Lagrange's Theorem and Applications
. 140
4.12
Euler's Center of an Inscribed Polygon
. 148
4.13
Some Geometric Transformations of the Complex Plane
. 151
4.13.1
Translation
. 151
4.13.2
Reflection in the real axis
. 152
4.13.3
Reflection in a point
. 152
4.13.4
Rotation
. 153
4.13.5
Isometric transformation of the complex plane
. 153
4.13.6
Morley's theorem
. 155
4.13.7
Homothecy
. 158
4.13.8
Problems
. 160
5
Olympiad-Caliber Problems
161
5.1
Problems Involving Moduli and Conjugates
. 161
5.2
Algebraic Equations and Polynomials
. 177
5.3
From Algebraic Identities to Geometric Properties
. 181
5.4
Solving Geometric Problems
. 190
5.5
Solving Trigonometric Problems
. 214
5.6
More on the ntb Roots of Unity
. 220
viii Contents
5.7 Problems
Involving
Polygons. 229
5.8
Complex Numbers and Combinatorics
. 237
5.9
Miscellaneous Problems
. 246
6
Answers, Hints and Solutions to Proposed Problems
253
6.1
Answers, Hints and Solutions to Routine Problems
. 253
6.1.1
Complex numbers in algebraic representation (pp.
18-21) . . . 253
6.1.2
Geometric interpretation of the algebraic operations (p.
27) . . 258
6.1.3
Polar representation of complex numbers (pp.
39-41). 258
6.1.4
The nlb roots of unity (p.
52). 260
6.1.5
Some geometric transformations of the complex plane (p.
160) 261
6.2
Solutions to the Olympiad-Caliber Problems
. 262
6.2.1
Problems involving moduli and conjugates (pp.
175-176) . . . 262
6.2.2
Algebraic equations and polynomials (p.
181). 269
6.2.3
From algebraic identities to geometric properties (p.
190) . . . 272
6.2.4
Solving geometric problems (pp.
211-213). 274
6.2.5
Solving trigonometric problems (p.
220). 287
6.2.6
More on the
и"1
roots of unity (pp.
228-229). 289
6.2.7
Problems involving polygons (p.
237) . 292
6.2.8
Complex numbers and combinatorics (p.
245). 298
6.2.9
Miscellaneous problems (p.
252). 302
Glossary
307
References
313
Index of Authors
317
Subject Index
319
Complex
Numbers
2,
It is impossible to imagine modern mathematics without complex
numbers. Complex Numbers from A to.
Z
introduces the reader
to this fascinating subject which, from the time of
Euler,
has
become one of the most utilized in mathematics.
The exposition concentrates on key concepts and then elemen¬
tary results concerning these numbers. The reader learns how
complex numbers can be used to solve algebraic equations and
to understand the geometric interpretation of complex numbers
and the operations involving them.
The theoretical parts of the book are augmented with rich exer¬
cises and problems at various levels of difficulty. A special fea¬
ture of the book is the last chapter, a selection of outstanding
Olympiad and other important mathematical contest problems
solved by employing the methods already presented.
The book reflects the unique experience of the authors. It distills
a vast mathematical literature, most of which is unknown to the
western public, and captures the essence of an abundant prob¬
lem culture. The target audience includes undergraduates, high
school students and their teachers, mathematical contestants
(such as those training for Olympiads or the W. L. Putnam
о
Mathematical Competition) and their coaches, as well as anyone
interested in essential mathematics. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Andreescu, Titu 1956- Andrica, Dorin 1956- |
| author_GND | (DE-588)12217805X (DE-588)113881789 |
| author_facet | Andreescu, Titu 1956- Andrica, Dorin 1956- |
| author_role | aut aut |
| author_sort | Andreescu, Titu 1956- |
| author_variant | t a ta d a da |
| building | Verbundindex |
| bvnumber | BV021294877 |
| callnumber-first | Q - Science |
| callnumber-label | QA255 |
| callnumber-raw | QA255 |
| callnumber-search | QA255 |
| callnumber-sort | QA 3255 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 130 SK 180 |
| classification_tum | MAT 100f |
| ctrlnum | (OCoLC)249567614 (DE-599)BVBBV021294877 |
| dewey-full | 512.788 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.788 |
| dewey-search | 512.788 |
| dewey-sort | 3512.788 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Greatly expanded and substantially enhanced version of the Romanian ed. |
| format | Book |
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| genre | Lehrbuch - Komplexe Zahl |
| genre_facet | Lehrbuch - Komplexe Zahl |
| id | DE-604.BV021294877 |
| illustrated | Illustrated |
| index_date | 2024-07-02T13:50:54Z |
| indexdate | 2024-07-09T20:34:57Z |
| institution | BVB |
| isbn | 0817643265 9780817643263 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014615658 |
| oclc_num | 249567614 |
| open_access_boolean | |
| owner | DE-824 DE-20 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-824 DE-20 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-11 |
| physical | XI, 321 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Birkhäuser |
| record_format | marc |
| spelling | Andreescu, Titu 1956- Verfasser (DE-588)12217805X aut Numere complexe de la A la ... Z Complex numbers from A to ... Z Titu Andreescu ; Dorin Andrica Greatly expanded and substantially enhanced version of the Romanian ed. Boston [u.a.] Birkhäuser 2006 XI, 321 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Numbers, Complex Komplexe Zahl (DE-588)4128698-4 gnd rswk-swf Lehrbuch - Komplexe Zahl Komplexe Zahl (DE-588)4128698-4 s DE-604 Andrica, Dorin 1956- Verfasser (DE-588)113881789 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014615658&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014615658&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Andreescu, Titu 1956- Andrica, Dorin 1956- Complex numbers from A to ... Z Numbers, Complex Komplexe Zahl (DE-588)4128698-4 gnd |
| subject_GND | (DE-588)4128698-4 |
| title | Complex numbers from A to ... Z |
| title_alt | Numere complexe de la A la ... Z |
| title_auth | Complex numbers from A to ... Z |
| title_exact_search | Complex numbers from A to ... Z |
| title_exact_search_txtP | Complex numbers from A to ... Z |
| title_full | Complex numbers from A to ... Z Titu Andreescu ; Dorin Andrica |
| title_fullStr | Complex numbers from A to ... Z Titu Andreescu ; Dorin Andrica |
| title_full_unstemmed | Complex numbers from A to ... Z Titu Andreescu ; Dorin Andrica |
| title_short | Complex numbers from A to ... Z |
| title_sort | complex numbers from a to z |
| topic | Numbers, Complex Komplexe Zahl (DE-588)4128698-4 gnd |
| topic_facet | Numbers, Complex Komplexe Zahl Lehrbuch - Komplexe Zahl |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014615658&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014615658&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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