What is 3.142857128571285712857128571... as a Fraction?

3.1428571 equals 314282579999900 as a fraction.

How do you turn 3.1428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.1428571 repeating into a fraction, begin writing this simple equation:

n = 3.1428571 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (28571) and 2 digits in the non-repeating part (14). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 31428571.28571 (equation 2)

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

10000000 × n = 31428571.28571
     100 × n = 3.1428571
 9999900 × n = 31428257

n = 314282579999900 (answer)

3.1428571 = 314282579999900 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 314282579999900 is also equal to 314285579999900 when expressed as a mixed number.

The repeating decimal 3.1428571 (vinculum notation) has a repeated block length of 5. It is also represented as 3.14285712857128571... (ellipsis notation) which equals approximately 3.142857128571285712857128571 (decimal approximation)(*).

The recurring decimal 3.1428571 can be written as a ratio of two integers having 31428257 as the numerator and 9999900 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.