Fractions are everywhere and yet most of us learn only basic and rather dry facts about fractions in primary school. This book makes fractions come to life in a friendly, lively, and accessible way, detailing the history of fractions and their crucial role in the work of mathematicians from various cultures throughout the ages. The book begins by outlining the importance of rational numbers and links ancient Babylonian mathematics with modern processes for determining their decimal expansions and the period length of repeating decimals, which are worked out in full. This then leads to the study of infinite sums, especially to geometric series and the notions of convergence and divergence. The text goes on to explain the importance of the Fibonacci numbers, as well as the Cantor set and the Sierpinski carpet. Much of elementary number theory is introduced including congruence classes, the Euler phi function, the Euclidean algorithm, and some Diophantine equations. The book also discusses many historical applications of fractions, including Christiaan Huygens's cogwheeled planetarium and Archimedes's approximation to the value of pi, as well as an extensive study of the importance of Egyptian fractions. Finally, it outlines modern applications of fractions, such as the fair apportionment of a cake, variations on slicing a pizza, probability questions involving markings on a stick, and ways to divide a bar of gold in order to pay wages for various numbers of days. Accessible to anyone with a passion for the history of mathematics who wishes to delve deeper into the wonderful world of fractions, this book will also be of special interest to teachers of mathematics and students of all ages.