commutative

/kə'mju:tətiv/

Thân thiện

In mathematics, addition is a commutative operation.

Definition

1. Adjective:
   • (Mathematics) Independent of order: Describing a binary operation (an operation that combines two elements) where changing the order of the operands does not change the result. For example, in the operation a + b, the result is the same as b + a.
   • (General) Relating to substitution or exchange: Pertaining to the act of replacing one thing with another or an exchange between parties. This meaning is less common in modern general usage but is the root of the mathematical term.

Usage Examples

• Adjective:
  • Addition and multiplication of real numbers are commutative operations.
  • The commutative property is a fundamental concept in elementary algebra.
  • Matrix multiplication is not commutative; A×B does not generally equal B×A.

Advanced Usage

• "commutative diagram": In category theory, a diagram of objects and morphisms where all directed paths with the same start and end points lead to the same result by composition, illustrating a form of commutativity.
  • The proof relied on showing that a certain square in the category was a commutative diagram.
• "commutative ring": In abstract algebra, a ring in which the multiplication operation is commutative.
  • The set of integers forms a commutative ring under standard addition and multiplication.

Variants and Related Words

• Commutativity (n): The property of being commutative.
  • The commutativity of addition is often taken for granted.
• Commutator (n): In mathematics and physics, an operator that measures the failure of commutativity for two operations.
  • In quantum mechanics, the commutator of position and momentum operators is central to the uncertainty principle.
• Noncommutative (adj): Not commutative.
  • The study of noncommutative geometry is a major area of modern mathematics.

Synonyms

• Order-independent: (For the mathematical sense) Not affected by the sequence of elements.
• Interchangeable: (For the general sense) Capable of being exchanged or substituted.

Antonyms

• Noncommutative: Not commutative; order-dependent.
• Anticommutative: A specific property where changing the order introduces a negative sign (e.g., the cross product of vectors: a × b = - (b × a)).

Học thuật

Adjective

1. (of a binary operation) independent of order; as in e.g. "a x b = b x a"