What is 2.14857142857142857142857142857142857142... as a Fraction?

2.14857142 equals 376175 as a fraction.

How do you turn 2.14857142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.14857142 repeating into a fraction, begin writing this simple equation:

n = 2.14857142 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (857142) and 2 digits in the non-repeating part (14). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 214857142.857142 (equation 2)

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

100000000 × n = 214857142.857142
      100 × n = 2.14857142
 99999900 × n = 214856928

21485692899999900 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 571428 (the GCF - greatest common factor).

n = 21485692899999900 = 214856928 ÷ 57142899999900 ÷ 571428 = 376175. So,

2.14857142 = 376175 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 376175 is also equal to 226175 when expressed as a mixed number.

The repeating decimal 2.14857142 (vinculum notation) has a repeated block length of 6. It is also represented as 2.14857142857142857142... (ellipsis notation) which equals approximately 2.14857142857142857142857142857142 (decimal approximation)(*).

The recurring decimal 2.14857142 can be written as a ratio of two integers having 376 as the numerator and 175 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.