The Real Numbers 1
The basic properties of the set of real numbersare given below.
Closure: For all real numbersthe sumand the productare real numbers.
Associative laws: For all real numbers
and
Commutative laws: For all real numbers
and
Distributive laws: For all real numbers
and
Identity elements: There are real numbersandsuch that for all real numbers
andand
and
Inverse elements: For each real numberthe equations
and
have solutionsin the set of real numbers, called the additive inverse of denoted by
For each nonzero real numberthe equationsand
have solutionsin the set of real numbers, called the multiplicative inverse ofdenoted by
Here are some additional properties of real numberswhich can be proved from the properties listed above.
Ifthen
Ifandis nonzero, then