What is 13.37142871428714287142871428... as a Fraction?

13.371428 equals 2674259199998 as a fraction.

How do you turn 13.371428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.371428 repeating into a fraction, begin writing this simple equation:

n = 13.371428 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (71428) and 1 digit in the non-repeating part (3). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 13371428.71428 (equation 2)

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

1000000 × n = 13371428.71428
     10 × n = 13.371428
 999990 × n = 13371295

13371295999990 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 5 (the GCF - greatest common factor).

n = 13371295999990 = 13371295 ÷ 5999990 ÷ 5 = 2674259199998. So,

13.371428 = 2674259199998 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 2674259199998 is also equal to 1374285199998 when expressed as a mixed number.

The repeating decimal 13.371428 (vinculum notation) has a repeated block length of 5. It is also represented as 13.3714287142871428... (ellipsis notation) which equals approximately 13.37142871428714287142871428 (decimal approximation)(*).

The recurring decimal 13.371428 can be written as a ratio of two integers having 2674259 as the numerator and 199998 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.