What is 1.287142871428714287142871428... as a Fraction?

1.2871428 equals 12871399999 as a fraction.

How do you turn 1.2871428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.2871428 repeating into a fraction, begin writing this simple equation:

n = 1.2871428 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (71428) and 2 digits in the non-repeating part (28). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 12871428.71428 (equation 2)

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

10000000 × n = 12871428.71428
     100 × n = 1.2871428
 9999900 × n = 12871300

128713009999900 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 100 (the GCF - greatest common factor).

n = 128713009999900 = 12871300 ÷ 1009999900 ÷ 100 = 12871399999. So,

1.2871428 = 12871399999 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 12871399999 is also equal to 12871499999 when expressed as a mixed number.

The repeating decimal 1.2871428 (vinculum notation) has a repeated block length of 5. It is also represented as 1.28714287142871428... (ellipsis notation) which equals approximately 1.287142871428714287142871428 (decimal approximation)(*).

The recurring decimal 1.2871428 can be written as a ratio of two integers having 128713 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.