A commutator is a mathematical concept used primarily in group theory and quantum mechanics. It measures the extent to which two operations, such as transformations or measurements, fail to commute, meaning the order in which they are applied matters. The commutator of two elements A and B is defined as [A, B] = AB - BA . If the commutator is zero, the operations can be performed in any order without affecting the outcome. In quantum mechanics, commutators play a crucial role in understanding the behavior of observables like position and momentum. For example, the position and momentum operators have a non-zero commutator, indicating that they cannot be simultaneously measured with arbitrary precision, a principle known as the Heisenberg uncertainty principle. This highlights the fundamental limits of measurement in the quantum realm.