Algebraic Properties
The addition (+) and multiplication (×) operations on natural numbers have several algebraic properties:
- Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.
- Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
- Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.
- Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.
- Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c)
- No zero divisors: if a and b are natural numbers such that a × b = 0 then a = 0 or b = 0.
Read more about this topic: Natural Number
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—Henry David Thoreau (1817–1862)
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—John Locke (1632–1704)