What is 14.5714257142571425714257142... as a Fraction?

14.57142 equals 145712899999 as a fraction.

How do you turn 14.57142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 14.57142 repeating into a fraction, begin writing this simple equation:

n = 14.57142 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (57142), so multiply both sides by 10⁵ = 100000.

100000 × n = 1457142.57142 (equation 2)

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

100000 × n = 1457142.57142
     1 × n = 14.57142
 99999 × n = 1457128

n = 145712899999 (answer)

14.57142 = 145712899999 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 145712899999 is also equal to 145714299999 when expressed as a mixed number.

The repeating decimal 14.57142 (vinculum notation) has a repeated block length of 5. It is also represented as 14.571425714257142... (ellipsis notation) which equals approximately 14.5714257142571425714257142 (decimal approximation)(*).

The recurring decimal 14.57142 can be written as a ratio of two integers having 1457128 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.