What is 14.25714257142571425714257142... as a Fraction?

14.257142 equals 142570099999 as a fraction.

How do you turn 14.257142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 14.257142 repeating into a fraction, begin writing this simple equation:

n = 14.257142 (equation 1)

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

1000000 × n = 14257142.57142 (equation 2)

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

1000000 × n = 14257142.57142
     10 × n = 14.257142
 999990 × n = 14257000

14257000999990 could be the answer, but it still can be put in the simplest form, i.e., reduced.

To simplify this fraction, divide the numerator and denominator by 10 (the GCF - greatest common factor).

n = 14257000999990 = 14257000 ÷ 10999990 ÷ 10 = 142570099999. So,

14.257142 = 142570099999 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 142570099999 is also equal to 142571499999 when expressed as a mixed number.

The repeating decimal 14.257142 (vinculum notation) has a repeated block length of 5. It is also represented as 14.2571425714257142... (ellipsis notation) which equals approximately 14.25714257142571425714257142 (decimal approximation)(*).

The recurring decimal 14.257142 can be written as a ratio of two integers having 1425700 as the numerator and 99999 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.