REAL NUMBERS, R

5. DECIMALS AND REAL NUMBERS

Every real number can be represented by a “non-terminating decimal”. The decimal representation of a rational number p/q can be found by “dividing the denominator q into the numerator p”. If the indicated division terminates, as

for

                 3/8 = .375

We write 3/8 = .375000

             Or 3/8 = .374999…

If we indicated division of q into p does not terminate, then it is known that a block of digits will continually be repeated; for example, 2/11 = .181818…

We now state the basic fact connecting decimals and real numbers. The rational numbers correspond precisely to those decimals in which a block of digits is continually repeated, and the irrational numbers correspond to the other non-terminating decimals.