What is 1.14285742857428574285742857... as a Fraction?

1.142857 equals 571423499995 as a fraction.

How do you turn 1.142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.142857 repeating into a fraction, begin writing this simple equation:

n = 1.142857 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (42857) and 1 digit in the non-repeating part (1). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 1142857.42857 (equation 2)

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

1000000 × n = 1142857.42857
     10 × n = 1.142857
 999990 × n = 1142846

1142846999990 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 2 (the GCF - greatest common factor).

n = 1142846999990 = 1142846 ÷ 2999990 ÷ 2 = 571423499995. So,

1.142857 = 571423499995 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 571423499995 is also equal to 171428499995 when expressed as a mixed number.

The repeating decimal 1.142857 (vinculum notation) has a repeated block length of 5. It is also represented as 1.1428574285742857... (ellipsis notation) which equals approximately 1.14285742857428574285742857 (decimal approximation)(*).

The recurring decimal 1.142857 can be written as a ratio of two integers having 571423 as the numerator and 499995 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.