## 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 numbersand

*Commutative laws:*For all real numbersand

*Distributive laws:*For all real numbersand

*Identity elements:*There are real numbersandsuch that for all real numbersandand

and

*Inverse elements:*For each real numberthe equationsand

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