What is 2.13857142857142857142857142857142857142... as a Fraction?

2.13857142 equals 1497700 as a fraction.

How do you turn 2.13857142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.13857142 repeating into a fraction, begin writing this simple equation:

n = 2.13857142 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (857142) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 213857142.857142 (equation 2)

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

100000000 × n = 213857142.857142
      100 × n = 2.13857142
 99999900 × n = 213856929

21385692999999900 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 142857 (the GCF - greatest common factor).

n = 21385692999999900 = 213856929 ÷ 14285799999900 ÷ 142857 = 1497700. So,

2.13857142 = 1497700 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 1497700 is also equal to 297700 when expressed as a mixed number.

The repeating decimal 2.13857142 (vinculum notation) has a repeated block length of 6. It is also represented as 2.13857142857142857142... (ellipsis notation) which equals approximately 2.13857142857142857142857142857142 (decimal approximation)(*).

The recurring decimal 2.13857142 can be written as a ratio of two integers having 1497 as the numerator and 700 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.