As the famous Pythagorean statement reads, 'Number rules the universe', and its veracity is proven in the many mathematical discoveries that have accelerated the development of science, engineering, and even philosophy. A so called "art of the mind", mathematics has guided and stimulated many aspects of human innovation down through the centuries.
In this book, Marcel Danesi presents a historical overview of the ten greatest achievements in mathematics, and dynamically explores their importance and effects on our daily lives. Considered as a chain of events rather than isolated incidents, Danesi takes us from the beginnings of modern day mathematics with Pythagoras, through the concept of zero, right the way up to modern computational algorithms.
Loaded with thought-provoking practical exercises and puzzles, Pythagoras' Legacy allows the reader to apply their knowledge and discover the significance of mathematics in their everyday lives.
Author(s): Marcel Danesi
Publisher: Oxford University Press, USA
Year: 2020
Language: English
Pages: x+171
Cover
Pythagoras’ Legacy: Mathematics in Ten Great Ideas
Copyright
Contents
Preface
Chapter 1: The Pythagorean theorem: The birth of mathematics
Prologue
The Pythagorean theorem
Proof
Discovery of √2
Practical uses
Pattern
Fermat’s Last Theorem
Epilogue
Explorations
1. Bhāskara’s snake and peacock puzzle
2. Abu al-Wafa’s assembly puzzle
3. Apollonius’ problem
4. Measurement problem
5. Gardner’s tricky triangle puzzle
Chapter 2: Prime numbers: The DNA of mathematics
Prologue
The infinity of primes
The Fundamental Theorem of Arithmetic
Searching for the primes
The Riemann Hypothesis
Epilogue
Explorations
1. Dudeney’s prime number magic square
2. Unravel the number
3. Prime number pattern
4. Twin primes
5. A prime number riddle
Chapter 3: Zero: Place-holder and peculiar number
Prologue
Negative numbers
Analytic geometry
Division by zero
The zero exponent
Binary digits
Epilogue
Explorations
1. A contradiction
2. Binary arithmetic
3. Classic snail problem
4. Firefighter puzzle
5. Trick problem
Chapter 4: π (Pi): A ubiquitous and strange number
Prologue
Value
Transcendental numbers
Manifestations
Epilogue
Explorations
1. A way to calculate π
2. A circular walk
3. A reverse puzzle
4. String around a circle
5. Pythagoras meets π
Chapter 5: Exponents: Notation and discovery
Prologue
Exponential notation
Exponential arithmetic
Pascal’s Triangle
Logarithms
Epilogue
Explorations
1. Square numbers
2. Exponential arithmetic
3. Generational logarithm
4. A tough nut
5. Laws of exponents
Chapter 6: e: A very special number
Prologue
Mathematical connectivity
Euler’s identity
Epilogue
Explorations
1. A possible series for e
2. Another series
3. Plotting a function
4. Compound interest
5. The exponential function ex
Chapter 7: i: Imaginary numbers
Prologue
Quadratic equations
Complex numbers
Fundamental Theorem of Algebra
Epilogue
Explorations
1. Imaginary numbers
2. A square root
3. Conjugates
4. Conjugate arithmetic
5. Powers of complex numbers
Chaptet 8: Infinity: A counterintuitive and paradoxical idea
Prologue
Zeno’s paradoxes
The Liar Paradox
Galileo’s and Cantor’s paradoxes
Hilbert’s infinite hotel paradox
Epilogue
Explorations
1. Einstein’s paradox
2. Cantorian method
3. Alexia’s paradox
4. Lateral thinking puzzles
5. Cantor again
Chapter 9: Decidability: The foundations of mathematics
Prologue
Consistency
Axiomatic structure
Undecidability
Epilogue
Explorations
1. Smullyan’s Gödelian puzzle
2. Gardner’s box logic puzzle
3. Dudeney’s logic puzzle
4. A derivative of Gardner’s puzzle
5. Deception logic
Chapter 10: The algorithm: Mathematics and computers
Prologue
Algorithms
Computability
Epilogue
Explorations
1. Measuring algorithm
2. Movement algorithm
3. Alcuin’s river crossing puzzle
4. A complex version
5. Possibility versus impossibility
Answers
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
References and Bibliography
Index
