Pure Mathematics studies the properties of abstract objects and concepts (such as numbers, functions, spaces, graphs and even reasoning itself), usually motivated by their own intrinsic interest or beauty rather than by specific applications outside of mathematics.
This is not to say that Pure Mathematics is not applicable indeed, it has a long history of making a fundamental impact on society. For example, the invention of complex numbers by Cardano in 16th century made possible the quantum mechanical vision of reality in 20th century, which in turn led to the development of semiconductors and modern computers, and hence the internet and our modern information-based society. The development of number theory and algebraic geometry up until the end of 19th century provided the level of cryptography needed to secure internet banking in the beginning of 21st century. The invention of Riemannian geometry during the 19th century provided the mathematical tools for Einstein's theory of general relativity, which describes space-time as an intrinsically curved four-dimensional object whose geometry is determined by the distribution of mass within it. More recently, discoveries in the theory of dynamical systems have considerable applications in other fields, such as biology, and we can expect further impact of pure mathematics on many areas of science and society in the future. In addition to eventual applications, the deepest problems studied by pure mathematics, such as the Riemann hypothesis or the Poincaré conjecture, shed light at the very basic principles of our world.
