Arno Berger is associate professor of mathematics at the University of Alberta. He is the author of Chaos and Chance: An Introduction to Stochastic Aspects of Dynamics. Theodore P. Hill is professor emeritus of mathematics at the Georgia Institute of Technology and research scholar in residence at the California Polytechnic State University.

"This is a marvelous and excellent introduction."--Adhemar Bultheel, European Mathematical Society Bulletin "A must-read for novices and experts alike. It can be used for a graduate-level topics course or as a reference text for researchers in the field. The exposition is outstanding, with hundreds of carefully chosen examples, figures and diagrams to illustrate the theory. For those who are up for a challenge, the book contains several open problems as well. An Introduction to Benford's Law will surely be the go-to text on the subject for years to come."--Pieter C. Allaart, Mathematical Reviews

• Preface vii 1 Introduction 1 1.1 History 3 1.2 Empirical evidence 4 1.3 Early explanations 6 1.4 Mathematical framework 7 2 Significant Digits and the Significand 11 2.1 Significant digits 11 2.2 The significand 12 2.3 The significand sigma-algebra 14 3 The Benford Property 22 3.1 Benford sequences 23 3.2 Benford functions 28 3.3 Benford distributions and random variables 29 4 The Uniform Distribution and Benford's Law 43 4.1 Uniform distribution characterization of Benford's law 43 4.2 Uniform distribution of sequences and functions 46 4.3 Uniform distribution of random variables 54 5 Scale-, Base-, and Sum-Invariance 63 5.1 The scale-invariance property 63 5.2 The base-invariance property 74 5.3 The sum-invariance property 80 6 Real-valued Deterministic Processes 90 6.1 Iteration of functions 90 6.2 Sequences with polynomial growth 93 6.3 Sequences with exponential growth 97 6.4 Sequences with super-exponential growth 101 6.5 An application to Newton's method 111 6.6 Time-varying systems 116 6.7 Chaotic systems: Two examples 124 6.8 Differential equations 127 7 Multi-dimensional Linear Processes 135 7.1 Linear processes, observables, and difference equations 135 7.2 Nonnegative matrices 139 7.3 General matrices 145 7.4 An application to Markov chains 162 7.5 Linear difference equations 165 7.6 Linear differential equations 170 8 Real-valued Random Processes 180 8.1 Convergence of random variables to Benford's law 180 8.2 Powers, products, and sums of random variables 182 8.3 Mixtures of distributions 202 8.4 Random maps 213 9 Finitely Additive Probability and Benford's Law 216 9.1 Finitely additive probabilities 217 9.2 Finitely additive Benford probabilities 219 10 Applications of Benford's Law 223 10.1 Fraud detection 224 10.2 Detection of natural phenomena 225 10.3 Diagnostics and design 226 10.4 Computations and Computer Science 228 10.5 Pedagogical tool 230 List of Symbols 231 Bibliography 234 Index 245

Arno Berger is associate professor of mathematics at the University of Alberta. He is the author of Chaos and Chance: An Introduction to Stochastic Aspects of Dynamics. Theodore P. Hill is professor emeritus of mathematics at the Georgia Institute of Technology and research scholar in residence at the California Polytechnic State University.

"This is a marvelous and excellent introduction."--Adhemar Bultheel, European Mathematical Society Bulletin "A must-read for novices and experts alike. It can be used for a graduate-level topics course or as a reference text for researchers in the field. The exposition is outstanding, with hundreds of carefully chosen examples, figures and diagrams to illustrate the theory. For those who are up for a challenge, the book contains several open problems as well. An Introduction to Benford's Law will surely be the go-to text on the subject for years to come."--Pieter C. Allaart, Mathematical Reviews

• Preface vii 1 Introduction 1 1.1 History 3 1.2 Empirical evidence 4 1.3 Early explanations 6 1.4 Mathematical framework 7 2 Significant Digits and the Significand 11 2.1 Significant digits 11 2.2 The significand 12 2.3 The significand sigma-algebra 14 3 The Benford Property 22 3.1 Benford sequences 23 3.2 Benford functions 28 3.3 Benford distributions and random variables 29 4 The Uniform Distribution and Benford's Law 43 4.1 Uniform distribution characterization of Benford's law 43 4.2 Uniform distribution of sequences and functions 46 4.3 Uniform distribution of random variables 54 5 Scale-, Base-, and Sum-Invariance 63 5.1 The scale-invariance property 63 5.2 The base-invariance property 74 5.3 The sum-invariance property 80 6 Real-valued Deterministic Processes 90 6.1 Iteration of functions 90 6.2 Sequences with polynomial growth 93 6.3 Sequences with exponential growth 97 6.4 Sequences with super-exponential growth 101 6.5 An application to Newton's method 111 6.6 Time-varying systems 116 6.7 Chaotic systems: Two examples 124 6.8 Differential equations 127 7 Multi-dimensional Linear Processes 135 7.1 Linear processes, observables, and difference equations 135 7.2 Nonnegative matrices 139 7.3 General matrices 145 7.4 An application to Markov chains 162 7.5 Linear difference equations 165 7.6 Linear differential equations 170 8 Real-valued Random Processes 180 8.1 Convergence of random variables to Benford's law 180 8.2 Powers, products, and sums of random variables 182 8.3 Mixtures of distributions 202 8.4 Random maps 213 9 Finitely Additive Probability and Benford's Law 216 9.1 Finitely additive probabilities 217 9.2 Finitely additive Benford probabilities 219 10 Applications of Benford's Law 223 10.1 Fraud detection 224 10.2 Detection of natural phenomena 225 10.3 Diagnostics and design 226 10.4 Computations and Computer Science 228 10.5 Pedagogical tool 230 List of Symbols 231 Bibliography 234 Index 245

Arno Berger is associate professor of mathematics at the University of Alberta. He is the author of Chaos and Chance: An Introduction to Stochastic Aspects of Dynamics. Theodore P. Hill is professor emeritus of mathematics at the Georgia Institute of Technology and research scholar in residence at the California Polytechnic State University.

"This is a marvelous and excellent introduction."--Adhemar Bultheel, European Mathematical Society Bulletin "A must-read for novices and experts alike. It can be used for a graduate-level topics course or as a reference text for researchers in the field. The exposition is outstanding, with hundreds of carefully chosen examples, figures and diagrams to illustrate the theory. For those who are up for a challenge, the book contains several open problems as well. An Introduction to Benford's Law will surely be the go-to text on the subject for years to come."--Pieter C. Allaart, Mathematical Reviews

• Preface vii 1 Introduction 1 1.1 History 3 1.2 Empirical evidence 4 1.3 Early explanations 6 1.4 Mathematical framework 7 2 Significant Digits and the Significand 11 2.1 Significant digits 11 2.2 The significand 12 2.3 The significand sigma-algebra 14 3 The Benford Property 22 3.1 Benford sequences 23 3.2 Benford functions 28 3.3 Benford distributions and random variables 29 4 The Uniform Distribution and Benford's Law 43 4.1 Uniform distribution characterization of Benford's law 43 4.2 Uniform distribution of sequences and functions 46 4.3 Uniform distribution of random variables 54 5 Scale-, Base-, and Sum-Invariance 63 5.1 The scale-invariance property 63 5.2 The base-invariance property 74 5.3 The sum-invariance property 80 6 Real-valued Deterministic Processes 90 6.1 Iteration of functions 90 6.2 Sequences with polynomial growth 93 6.3 Sequences with exponential growth 97 6.4 Sequences with super-exponential growth 101 6.5 An application to Newton's method 111 6.6 Time-varying systems 116 6.7 Chaotic systems: Two examples 124 6.8 Differential equations 127 7 Multi-dimensional Linear Processes 135 7.1 Linear processes, observables, and difference equations 135 7.2 Nonnegative matrices 139 7.3 General matrices 145 7.4 An application to Markov chains 162 7.5 Linear difference equations 165 7.6 Linear differential equations 170 8 Real-valued Random Processes 180 8.1 Convergence of random variables to Benford's law 180 8.2 Powers, products, and sums of random variables 182 8.3 Mixtures of distributions 202 8.4 Random maps 213 9 Finitely Additive Probability and Benford's Law 216 9.1 Finitely additive probabilities 217 9.2 Finitely additive Benford probabilities 219 10 Applications of Benford's Law 223 10.1 Fraud detection 224 10.2 Detection of natural phenomena 225 10.3 Diagnostics and design 226 10.4 Computations and Computer Science 228 10.5 Pedagogical tool 230 List of Symbols 231 Bibliography 234 Index 245