Properties of addition and multiplication on integers
Addition
Multiplication
Closure:
a + b is an integer
a × b is an integer
Associativity:
a + (b + c) = (a + b) + c
a × (b × c) = (a × b) × c
Commutativity:
a + b = b + a
a × b = b × a
Existence of an identity element:
a + 0 = a
a × 1 = a
Existence of inverse elements:
a + (−a) = 0
An inverse element usually does not exist at all.
Distributivity:
a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)
No zero divisors: (*)
If a × b = 0, then a = 0 or b = 0 (or both)
