In der folgenden Ausarbeitung werden einige zentrale mathematische Probleme und ihre Lösungsansätze beschrieben, u.a.: Zahlensysteme, Ungelöste Probleme der Antike, Befreundete Zahlen und pythagoreische Triple, Vollkommene Zahlen, Der Goldene Schnitt, Wahrscheinlichkeitsrechnung nach Pascal, Fibonacci-Zahlen, Goldbach-Vermutung, Primzahlen, Möbius-Funktion.
Table of Contents
I. The Ishango bone:
II. The calculating system of the Babylonians
III. The unsolvable mathematical problems of the ancient world:
The doubling of the cube:
The angle trisection:
Squaring the circle:
IV. The five Platonic solids:
(1) Tetrahedron:
(2) Hexahedron:
(3) Octahedron:
(4) Icosahedron:
(5) Dodecahedron:
V. The Pythagoreans: Amicable numbers
VI. Pythagorean triples:
VII. Euclid of Alexandria on perfect numbers:
VIII. Euclid’s Golden Ratio:
IX. The Sand Calculator:
X. The Bakhshali-Manuscript:
XI. The Indians and the number cipher:
XII. Indian-Arabian number system:
XIII. Ibn Al-Haitham (965 - 1039) on cubic numbers:
XIV. Abu Ali Al-Husain Ibn Sina (980 - 1037) on perfect numbers:
XV. Cardano’s „Ars Magna“:
XVI. Blaise Pascal’s probability calculation
XVII. John Napier (1550 - 1617) Logarithm:
XVIII. Kepler on Fibonacci numbers
XIX. Kepler on the relationship between the Golden Ration and the Fibonacci numbers
XX. Bernoulli’s inequation
XXI. Christian Goldbach’s conjecture
XXII. The Goldbach Conjecture:
XXIII. The Euler number e:
XXIV. The Stirling Formula (here also Euler number e is needed):
XXV. Gauß (and others) on Prime Numbers
XXVI. Gauß’ Prime Number Conjecture
XXVII. Lagrange Theorems:
XXVIII. Joseph-Louis Lagrange: Binominal Theorem
XXIX. Lagrange Theorems: The 4-Squares-Theorem:
XXX. Jurij Vega (I): Logarithms
XXXI: Jurij Vega (II):
XXXII. Jurij Vega (III):
XXXIII. The Möbius Function:
XXXIV: Srinivasa Ramanujan (1887-1920):
XXXV: The Italian mathematician Giuseppe Peano:
XXXVI: The Catalan Conjecture:
XXXVII. Cantor-Conjecture:
XXXVIII. The Conway Sequence:
XXXIX. The Brun Constant:
XL. The Birthday Paradox:
XLI. Paul Stäckel and the Twin Primes:
XLII. The ABC-Conjecture:
XLIII. The Andrica Conjecture:
Objectives & Core Topics
The primary objective of this text is to provide a concise historical overview of significant mathematical discoveries, problems, and the key figures behind them, spanning from ancient artifacts to modern conjectures. The document aims to illustrate the evolution of mathematical thought through specific theorems, number systems, and paradoxes.
- Historical evolution of counting systems and number representation (from Babylonians to modern numerals).
- In-depth exploration of classical unsolvable problems and famous mathematical conjectures.
- Biography-focused insights into influential mathematicians such as Euler, Pascal, Lagrange, and Ramanujan.
- Applications of probability theory, number theory, and logarithmic calculations.
Excerpt from the Book
Blaise Pascal’s probability calculation
Blaise Pascal, like other philosophers and mathematicians, was interested in the possibility of predicting future events.
Pascal was born in Clermont and lived from 1623 to 1662.
His central work is the Pensées, which was first published under the title of Pensées sur la religion et autres sujets in 1670, eight years after Pascal’s death.
In the second chapter of this book Pascal deals with the dialectic of gambits, later on also called “Pascal’s Gambit”.
Already at the age of twelve Pascal had published a much quoted work on conic sections.
Further much quoted works were: Potestatum numericarum summa, De numeris multiplicibus.
Pascal’s first interest in probability calculation arose when a close friend, Chevalier de Marre, who was addicted to gambling, asked Pascal to help him improve his chances in roulette.
Summary of Chapters
I. The Ishango bone: Discusses the 20,000-year-old Ishango bone and its notches, which represent early evidence of doubling, bisection, and prime numbers.
II. The calculating system of the Babylonians: Explains the Babylonian sexagesimal (base 60) positional number system and its enduring influence on modern time and angular measurement.
III. The unsolvable mathematical problems of the ancient world: Examines three classical geometric challenges: doubling the cube, trisecting an angle, and squaring the circle.
XVI. Blaise Pascal’s probability calculation: Details Pascal's contributions to probability theory and his philosophical reflections on God through a game-theoretic approach.
XXXIV: Srinivasa Ramanujan (1887-1920): Highlights the life and brilliance of Ramanujan, specifically focusing on the properties of the number 1729, known as the Hardy-Ramanujan number.
Keywords
Mathematics, Prime numbers, Fibonacci sequence, Golden ratio, Probability theory, Babylonians, Logarithms, Perfect numbers, Conjecture, Euler, Pascal, Ramanujan, Axioms, Geometry, Number systems
Frequently Asked Questions
What is the core focus of this work?
This work explores the historical development of mathematics, focusing on famous problems, key historical figures, and significant mathematical constants or conjectures throughout history.
What are the primary themes discussed?
The core themes include ancient numbering systems, the evolution of arithmetic and algebra, probability theory, prime number theory, and the contributions of legendary mathematicians.
What is the primary goal of this publication?
The goal is to inform readers about the lineage of mathematical thought and the specific breakthroughs that shaped the field from antiquity to the 20th century.
Which scientific methods are primarily utilized in the text?
The text employs historical analysis and didactic explanation of mathematical proofs, sequences, and formulas to demonstrate the logic used by historical mathematicians.
What content is covered in the main section?
The main sections cover specific topics such as the Ishango bone, Platonic solids, number systems (Babylonian, Indian-Arabian), perfect numbers, and various famous conjectures like the Goldbach or Catalan conjectures.
Which keywords best characterize this work?
Mathematics, prime numbers, number theory, historical mathematicians, and scientific conjectures are the defining terms of the text.
How does the text explain the significance of the number 1729?
Ramanujan famously identified 1729 as the smallest number expressible as the sum of two cubes in two different ways, challenging the notion that it was a "boring" number.
What is the "Birthday Paradox" mentioned in the text?
The Birthday Paradox demonstrates that in a group of only 23 people, there is a greater than 50% probability that at least two people share the same birthday.
How is the Babylonian number system relevant today?
The Babylonian base-60 system survives in our modern measurement of time (60 seconds, 60 minutes) and the division of a circle into 360 degrees.
- Quote paper
- Prof. Dr. phil. Frank Kostrzewa (Author), 2014, Famous mathematicians and their problems, Munich, GRIN Verlag, https://www.grin.com/document/282774
