Sets, such as numbers, can have two operations, such as addition and multiplication {algebra, system} {formal system, algebra}. Operations map elements, or element pairs, to elements: 1 + 2 = 3. Arithmetic is algebra. Elements are real numbers, and operations are addition and multiplication.
In arithmetic, a + (b + c) = (a + b) + c {associative law of addition}.
In arithmetic, a * (b * c) = (a * b) * c {associative law of multiplication}.
In arithmetic, a + b = b + a {commutative law of addition}.
In arithmetic, a * b = b * a {commutative law of multiplication}.
In arithmetic, a * (b + c) = a*b + a*c {distributive law}.
A number {identity element, addition} can sum with another number to make second number. A number {identity element, multiplication} can multiply with another number to make second number. In arithmetic, 0 is additive identity and 1 is multiplicative identity.
In arithmetic, numbers has numbers {additive inverse} {inverse element, algebra} that can add to make additive identity element: n + -n = 0. Numbers have numbers {multiplicative inverse} that can multiply to make multiplicative-identity element: n * (1/n) = 1.
In non-associative algebras, association-axiom replacement {Jacobi identity} can be (a, (b, c)) + (b, (c, a)) + (c, (a, b)) = 0. Commutative-axiom replacement can be (a, b) = -(b, a).
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Date Modified: 2022.0225