What is 13.15714257142571425714257142... as a Fraction?

13.157142 equals 13157011999990 as a fraction.

How do you turn 13.157142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.157142 repeating into a fraction, begin writing this simple equation:

n = 13.157142 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (57142) and 1 digit in the non-repeating part (1). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 13157142.57142 (equation 2)

Step 3: Now subtract equation 1 from equation 2 to cancel the repeating block (or repetend) out.

1000000 × n = 13157142.57142
     10 × n = 13.157142
 999990 × n = 13157011

n = 13157011999990 (answer)

13.157142 = 13157011999990 as the lowest possible fraction.

As the numerator is greater than the denominator, we have an improper fraction, so we can also express it as a mixed number, thus 13157011999990 is also equal to 13157141999990 when expressed as a mixed number.

The repeating decimal 13.157142 (vinculum notation) has a repeated block length of 5. It is also represented as 13.1571425714257142... (ellipsis notation) which equals approximately 13.15714257142571425714257142 (decimal approximation)(*).

The recurring decimal 13.157142 can be written as a ratio of two integers having 13157011 as the numerator and 999990 as the denominator. So, it is a rational number (named after ratio). It can be shown that a number is rational if its decimal representation is repeating or terminating.

(*) At present, there is no single universally accepted notation or phrasing for repeating decimals.