What is 2.6142857142857142857142857142857142857142857... as a Fraction?

2.6142857 equals 18370 as a fraction.

How do you turn 2.6142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.6142857 repeating into a fraction, begin writing this simple equation:

n = 2.6142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857) and 1 digit in the non-repeating part (6). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 26142857.142857 (equation 2)

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

10000000 × n = 26142857.142857
      10 × n = 2.6142857
 9999990 × n = 26142831

261428319999990 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 = 261428319999990 = 26142831 ÷ 1428579999990 ÷ 142857 = 18370. So,

2.6142857 = 18370 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 18370 is also equal to 24370 when expressed as a mixed number.

The repeating decimal 2.6142857 (vinculum notation) has a repeated block length of 6. It is also represented as 2.6142857142857142857... (ellipsis notation) which equals approximately 2.6142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 2.6142857 can be written as a ratio of two integers having 183 as the numerator and 70 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.