What is 0.4142857142857142857142857142857142857142857... as a Fraction?

0.4142857 equals 2970 as a fraction.

How do you turn 0.4142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.4142857 repeating into a fraction, begin writing this simple equation:

n = 0.4142857 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (142857) and 1 digit in the non-repeating part (4). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 4142857.142857 (equation 2)

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

10000000 × n = 4142857.142857
      10 × n = 0.4142857
 9999990 × n = 4142853

41428539999990 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 = 41428539999990 = 4142853 ÷ 1428579999990 ÷ 142857 = 2970. So,

0.4142857 = 2970 as the lowest possible fraction.

The repeating decimal 0.4142857 (vinculum notation) has a repeated block length of 6. It is also represented as 0.4142857142857142857... (ellipsis notation) which equals approximately 0.4142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 0.4142857 can be written as a ratio of two integers having 29 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.