The Real Numbers 1

The basic properties of the set of real numbersare given below.

1. Closure: For all real numbersthe sumand the productare real numbers.

2. Associative laws: For all real numbers

   and

3. Commutative laws: For all real numbers

   and

4. Distributive laws: For all real numbers

   and

5. Identity elements: There are real numbersandsuch that for all real numbers

   andand

   and

6. 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 equations

   and

   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

• 

• 

•