What is 0.7142142142... as a Fraction?

0.7142 equals 14271998 as a fraction.

How do you turn 0.7142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.7142 repeating into a fraction, begin writing this simple equation:

n = 0.7142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 1 digit in the non-repeating part (7). Multiply both sides by 10⁴ = 10000.

10000 × n = 7142.142 (equation 2)

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

10000 × n = 7142.142
   10 × n = 0.7142
 9990 × n = 7135

71359990 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 = 71359990 = 7135 ÷ 59990 ÷ 5 = 14271998. So,

0.7142 = 14271998 as the lowest possible fraction.

The repeating decimal 0.7142 (vinculum notation) has a repeated block length of 3. It is also represented as 0.7142142142... (ellipsis notation) which equals approximately 0.7142142142142142 (decimal approximation)(*).

The recurring decimal 0.7142 can be written as a ratio of two integers having 1427 as the numerator and 1998 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.