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

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