What is 13.7142714271427142... as a Fraction?

13.7142 equals 1371299999 as a fraction.

How do you turn 13.7142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.7142 repeating into a fraction, begin writing this simple equation:

n = 13.7142 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (7142), so multiply both sides by 10⁴ = 10000.

10000 × n = 137142.7142 (equation 2)

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

10000 × n = 137142.7142
    1 × n = 13.7142
 9999 × n = 137129

n = 1371299999 (answer)

13.7142 = 1371299999 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 1371299999 is also equal to 1371429999 when expressed as a mixed number.

The repeating decimal 13.7142 (vinculum notation) has a repeated block length of 4. It is also represented as 13.714271427142... (ellipsis notation) which equals approximately 13.71427142714271427142 (decimal approximation)(*).

The recurring decimal 13.7142 can be written as a ratio of two integers having 137129 as the numerator and 9999 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.