This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law's colorful history, rapidly growing body of empirical evidence, and wide range of applications.An Introduction to Benford's Law begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford's law.Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The text includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. This text can serve as both a primary reference and a basis for seminars and courses.

Preface vii1 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 s-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