Friday, 12 February 2016

decimal expansion - Rational number with more than ten digits



Could we have a repeating rational decimal number with more than 10 repeating digits (something like 0.0123456789801234567898...) after the decimal point?




What is the maximum number of repeating digits after the decimal point in a number?



Could the answer be generalized to state that we could / couldn’t have a repeating rational number in base b with more than b repeating digits?


Answer



The period of a periodic sequence of digits can be as large as you like. To see this, multiply the number by 10T, where T is the period, and then subtract the original number. Since this is definitely a whole number n – the repeating parts of the sequence cancel out – the original must have been a rational number, specifically n/(10T1).


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...