What is 0.255714571457145714... as a Fraction?

0.255714 equals 255689999900 as a fraction.

How do you turn 0.255714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.255714 repeating into a fraction, begin writing this simple equation:

n = 0.255714 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (5714) and 2 digits in the non-repeating part (25). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 255714.5714 (equation 2)

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

1000000 × n = 255714.5714
    100 × n = 0.255714
 999900 × n = 255689

n = 255689999900 (answer)

0.255714 = 255689999900 as the lowest possible fraction.

The repeating decimal 0.255714 (vinculum notation) has a repeated block length of 4. It is also represented as 0.25571457145714... (ellipsis notation) which equals approximately 0.2557145714571457145714 (decimal approximation)(*).

The recurring decimal 0.255714 can be written as a ratio of two integers having 255689 as the numerator and 999900 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.