Answer to Question #208530 in Real Analysis for Msa

Question

”A real number is rational if and only if it has a periodic decimal expansion.” Define the present usage of the word periodic and prove the statement.

Answer 1

Periodic means that the certain set of numbers are repeating after regular time intervals.

Proof:-

Since p⇔q is equivalent, logically, to (¬p)⇔(¬q) ,

 we instead prove that a real number is rational if and only if it has a repeating or terminating decimal] expansion.

Suppose a real number has a repeating or terminating decimal expansion. If it is terminating, then it is of the form  

a_0.a_1a_2a_3…a_n , which is equal to the rational number $\dfrac{a_0a_1a_2…a_n}{10^n}$ . If it is repeating, then it is of the form $a_0.a_1a_2…a_mb_1b_2…b_nb_1b_2…b_n$ … , which is equal to the rational number $\dfrac{a_0a_1a_2…a_m}{10^m}+\dfrac{b_1b_2…b_n}{10^{m+n}−10^m} .$

Conversely, suppose we have a rational number $\dfrac{p}{q}$ . Perform the division of p.00000… by q using the long division algorithm. Every step of the division will have one of q remainders. If the remainder eventually becomes zero, then the algorithm stops and the rational number’s decimal expansion terminates. If the remainder never becomes zero, then the same remainder will eventually reappear after at most (q−1) steps, in which case the same string of decimal digits (at most q−1 digits long) will appear over and over. Thus $\dfrac{p}{q}$  will have a repeating decimal expansion, as required.