This book covers the theory of the commutator and of non-commutative differentiation. This material is essential to classical electromagnetism, particle physics, and quantum field theory. Written by an author famous for his clear exposition, the material is clearly presented using no more than matrices and elementary differentiation. This book will be easily understood by anyone with a first year course in maths or physics. We are guided through the fundamentals of finite groups and how they relate to algebras like the complex numbers and the quaternions. We then discover that the commutator is a third algebraic opertion, and this leads to the differential within non-commutative algebras like the quaternions. The next thing we know is that our 4-dimensional space-time and electro-weak theory are falling out of simple mathematics. This book is a must read for all physicists and mathematicians.