Pythagoras' Legacy: Mathematics in Ten Great Ideas

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