The greatest innovation in mathematics was the Hindu-Arabic number system. Developed by Hindu mathematicians in India around 500 to 700 AD and studied by Arabic scholars, it was introduced to the West by Leonardo of Pisa in his 1202 manuscript Liber Abaci. Leonardo realised that the simple idea of representing positive whole numbers (and hence fractions as well) with a positional system using the ten basic symbols 0, 1, ..., 9, and manipulating them according to simple symbolic rules, would make it possible for any person to do their own arithmetic calculations. Numbers would no longer be the domain of an exclusive group of experts.
The widespread adoption of the new system in thirteenth century Italy not only provided the foundation for modern commerce and finance, it also paved the way for the development over the ensuing two centuries of quantitatively-based modern science by Galileo and others. An analogous example of such a life-changing democratising innovation in our own time is the development of the personal computer, which put computational and communicative power in the hands of ordinary people (and also made possible new advances in science).
Keith J Devlin professor of mathematics at Stanford University, executive director of the Center for the Study of Language and Information and a co-founder of Media X.