Why the repeating decimal is rational number?

Answer 1

A decimal expression is a rational number if it can be represented as a ratio of two integers, p/q where q > 0.

Suppose there are c digits after the decimal place where the digits are non-repeating, and then you get a repeating pattern of a string of d digits. Then the numerator of the rational form is the old original string including one lot of the repeated digits minus the original string with none of the repeating digits.

The denominator of the rational form is 10^c*(10^d - 1), which is a string of d 9s followed by c 0s.

For example

123.26159159… There are 2 digits, "26", after the decimal point before the repeats kick in so c = 2, and the repeating string "159" is 3 digits long so d = 3.

So the numerator is 12326159 – 12326 = 12313833

and the denominator is 99900.

Therefore the fraction is 12313833/99900.