Question

Express tha following number into decimals
327/507

Answer 1

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Every rational number has a terminating or a recurring i.e. repeating decimal representation. As the given number 327/507 = 109/169 is rational, and the denominator is 169 =13×13, its decimal expansion must be repeating, because only those rational numbers have terminating decimal representations whose denominator has no prime factors other than 2 and 5. Apparently the recurring period of this particular number is rather large. Non-terminating and nonrepeating decimals are irrational, which is not the case here.

P.S.: You can also prove the above result as follows: When you carry out the division after the decimal, each time putting a new 0 on the right, the remainders are all < the divider, which in this case is 169. So in at most 169 steps, you will either get a remainder 0, in which case it is a terminating decimal or else some earlier remainder will appear again. You may stop there, because from here onwards, the finite number of digits between the reappearing digits (with only one end of the two appearances included) will be repeating endlessly, and so it will be a recurring decimal.