Prove that any number with a recurring decimal expansion is equal to a fraction

Apr 2010
30
0
Can anyone prove that any number with a recurring decimal expansion is equal to a fraction?
 

simplependulum

MHF Hall of Honor
Jan 2009
715
427
Try to prove that if it is \(\displaystyle 0.a_1 a_2 a_3 ... a_n a_1 a_2 a_3 ... a_n a_1 a_2 a_3 ... a_n ... \)then it can be writen as a fraction : \(\displaystyle \frac{ a_1 a_2 a_3 ... a_n}{999...9} \) or \(\displaystyle \frac{ a_1 a_2 a_3 ... a_n}{ 10^n - 1 } \)