A marvelous compendium of mathematical symbols and their fascinating histories
Galileo famously wrote that the book of nature is written in mathematical language. The Language of Mathematics is a wide-ranging and beautifully illustrated collection of short, colorful histories of the most commonly used symbols in mathematics, providing readers with an engaging introduction to the origins, evolution, and conceptual meaning of each one.
In dozens of lively and informative entries, Raúl Rojas shows how today's mathematics stands on the shoulders of giants, mathematicians from around the world who developed mathematical notation through centuries of collective effort. He tells the stories of such figures as al-Khwārizmī, René Descartes, Joseph-Louis Lagrange, Carl Friedrich Gauss, Augustin-Louis Cauchy, Karl Weierstrass, Sofia Kovalevskaya, David Hilbert, and Kenneth Iverson. Topics range from numbers and variables to sets and functions, constants, and combinatorics. Rojas describes the mathematical problems associated with different symbols and reveals how mathematical notation has sometimes been an accidental process. The entries are self-contained and can be read in any order, each one examining one or two symbols, their history, and the variants they may have had over time.
An essential companion for math enthusiasts, The Language of Mathematics shows how mathematics is a living and evolving entity, forever searching for the best symbolism to express relationships between abstract concepts and to convey meaning.
| Acknowledgments | ix | ||
| Symbols and Expressions | xi | ||
| Introduction | 1 | ||
| I | The Long History of Mathematical Symbols | 5 | |
| The Birth of Algebra | 5 | ||
| How Do We Use Mathematical Symbols? | 10 | ||
| The Most Beautiful Mathematical Formulas | 14 | ||
| Why Do We Extract Roots? | 17 | ||
| II | Numbers and Variables | 22 | |
| Hindu-Arabic Numerals and Mercantilism | 22 | ||
| The Greek Alphabet and Its Predecessors | 27 | ||
| Zero | 32 | ||
| Symmetry of Symbols | 39 | ||
| The Variable x | 42 | ||
| Absolute Value and the Father of Analysis | 51 | ||
| Powers as Superscripts | 55 | ||
| Subscripts | 60 | ||
| The Decimal Point | 63 | ||
| III | Arithmetic Operators | 69 | |
| The Greek Cross of Addition | 69 | ||
| Subtraction and Negative Numbers | 73 | ||
| According to Adam Ries | 78 | ||
| The Cross of Multiplication | 82 | ||
| The Division Bar | 86 | ||
| Homer, the Obelus, and Division | 89 | ||
| IV | Relational and Grouping Operators | 94 | |
| No Two Things Can Be More Equal | 94 | ||
| Inequality Symbols | 97 | ||
| The Parenthesis versus the Vinculum | 100 | ||
| The Comma and the Period | 104 | ||
| V | Calculus/Analysis | 107 | |
| Star Wars: Leibniz versus Newton | 107 | ||
| The Partial Derivative | 113 | ||
| Nabla, the Harp of Assyria | 118 | ||
| John Wallis and Infinity | 122 | ||
| Delta | 125 | ||
| The Notation f(x) and the Concept of a Function | 128 | ||
| Epsilons, Deltas, and the Invention of Real Numbers | 134 | ||
| Reaching the Limit | 138 | ||
| The Mathematical Dart | 143 | ||
| VI | Sets and Functions | 149 | |
| The Existential Quantifier | 149 | ||
| The Universal Quantifier | 153 | ||
| ∈ Is for Belonging | 157 | ||
| The Set of Rational Numbers | 159 | ||
| Mathematics and Nothingness | 164 | ||
| Union and Intersection | 171 | ||
| The Aleph and the Paradise of Infinities | 173 | ||
| VII | Constants | 180 | |
| The Power of Imagination | 180 | ||
| Pi: The Archimedean Constant and the Ludolphian Number | 183 | ||
| Euler's Number and Exponential Growth | 189 | ||
| Planck's Constant: The Quantum of Action | 194 | ||
| The Speed of Light c | 196 | ||
| VIII | Combinatorics | 201 | |
| The Factorial | 201 | ||
| Sigma: Notation to Sink Your Teeth Into | 204 | ||
| A Floor and a Ceiling for Numbers | 207 | ||
| The Binomial Symbol | 209 | ||
| IX | Various Topics | 213 | |
| The Invisible Symbol: Einstein's Convention | 213 | ||
| The Halmos Box | 215 | ||
| Sine of Theta and Trigonometry | 218 | ||
| The Symbol of Congruence and Arithmetic in Miniature | 221 | ||
| Matrices: The Mother Structure | 224 | ||
| Publish or Perish: The First Scientific Journals | 228 | ||
| Epilogue | 235 | ||
| Bibliography | 237 | ||
| Index | 247 |
"Galileo said that the universe is written in the language of mathematics, but what is the language of mathematics? This entertaining book gives a rich historical answer: the often accidental, sometimes contradictory, and all too human story of mathematical symbolism and its evolution." --John Stillwell, author of The Story of Proof
" The Language of Mathematics is built on the inspired idea that you can generate a thorough and lively history of mathematics by telling the story of its symbols--the x, ∫, and the like that give a page of mathematical text its special color." --Jordan Ellenberg, author of Shape
"Rojas's robust and extensive history examines a broad array of mathematical symbols, pleasantly enriching the narration by situating each case within its cultural context and the evolution of mathematics as a discipline and language." --Teresa Rojano, Center for Research and Advanced Studies, National Polytechnic Institute, Mexico
"Rojas provides an engaging discussion of the history of mathematical notation and symbols and the mathematicians who invented them. A great read." --Oscar E. Fernandez, author of Calculus Simplified
"Beautifully written for all readers, The Language of Mathematics provides a great journey through the origins and cultural evolution of some of the most important mathematical symbols. Rojas deftly intertwines his historical analysis of the symbols with an engaging exploration of the mathematical concepts they represent." --Érika B. Roldán Roa, Max Planck Institute for Mathematics in the Sciences and ScaDS.AI, Leipzig University
"Offers the numeral nerds among us a guide to the fascinating history of mathematical symbols, some of which have become commonplace in everyday life." ---Brianne Kane, Scientific American
"I loved it." ---Jeff O'Neal, Book Riot: The Podcast
"A Choice Outstanding Academic Title of the Year"
Raúl Rojas is professor of mathematics and statistics at the University of Nevada, Reno, and professor emeritus of computer science and mathematics at the Free University of Berlin. A world-renowned expert in artificial intelligence, he is the author of the seminal book Neural Networks and the editor (with Ulf Hashagen) of The First Computers.