What does noncommutative mean?
Noncommutative describes an operation or mathematical structure for which order matters. If the order of two operations is reversed, the result may be different: for example, multiplying matrices A and B does not generally produce the same product as multiplying B and A. This contrasts with commutative operations, such as ordinary addition, for which order has no effect. The term is important in abstract algebra, ring theory, matrix mathematics, quantum mechanics, and computer science. Although it may describe any process in which sequence affects an outcome, its usual applications are technical. The word is closely related to commute and commutative and is commonly paired with terms such as operation, multiplication, algebra, and ring.
adjectiveDenoting an operation, structure, or system in which the order of operations matters, so that changing the sequence can change the result.
- Denoting an operation, structure, or system in which order matters and reversing the order of operations can change the result.
"Matrix multiplication is noncommutative: multiplying two matrices in one order need not give the same result as multiplying them in the reverse order."
"The theory relies heavily on noncommutative multiplication."
"Unlike addition, matrix multiplication is noncommutative, so changing the order of the matrices can change the product."
Noncommutative mathematics shows that changing the order of steps can change the outcome—an idea that also reshaped the foundations of quantum physics.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Where does noncommutative come from?
The word is formed from the prefix non- and the older adjective commutative. Commutative comes ultimately from Latin commutare, a form built from com- and mutare, both associated with changing or exchanging. The modern mathematical sense developed as noncommutative became a standard contrast with commutative in mathematical terminology.
Related word forms
How noncommutative is actually used
Chiefly formal and technical, especially in mathematics, algebra, physics, and computer science. The spelling "non-commutative" is also used, particularly in formal mathematical writing.
Easily confused with noncommutative
A commutative operation gives the same result regardless of the order of its inputs, whereas a noncommutative operation can give a different result when the order is reversed.