What is 3.2142857142857142857142857142857142857142857... as a Fraction?

3.2142857 equals 4514 as a fraction.

How do you turn 3.2142857 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.2142857 repeating into a fraction, begin writing this simple equation:

n = 3.2142857 (equation 1)

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

10000000 × n = 32142857.142857 (equation 2)

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

10000000 × n = 32142857.142857
      10 × n = 3.2142857
 9999990 × n = 32142825

321428259999990 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 714285 (the GCF - greatest common factor).

n = 321428259999990 = 32142825 ÷ 7142859999990 ÷ 714285 = 4514. So,

3.2142857 = 4514 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 4514 is also equal to 3314 when expressed as a mixed number.

The repeating decimal 3.2142857 (vinculum notation) has a repeated block length of 6. It is also represented as 3.2142857142857142857... (ellipsis notation) which equals approximately 3.2142857142857142857142857142857 (decimal approximation)(*).

The recurring decimal 3.2142857 can be written as a ratio of two integers having 45 as the numerator and 14 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.