What is 6.428571428571428571428571428571428571... as a Fraction?

6.428571 equals 457 as a fraction.

How do you turn 6.428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 6.428571 repeating into a fraction, begin writing this simple equation:

n = 6.428571 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (428571), so multiply both sides by 10⁶ = 1000000.

1000000 × n = 6428571.428571 (equation 2)

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

1000000 × n = 6428571.428571
      1 × n = 6.428571
 999999 × n = 6428565

6428565999999 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 = 6428565999999 = 6428565 ÷ 142857999999 ÷ 142857 = 457. So,

6.428571 = 457 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 457 is also equal to 637 when expressed as a mixed number.

The repeating decimal 6.428571 (vinculum notation) has a repeated block length of 6. It is also represented as 6.428571428571428571... (ellipsis notation) which equals approximately 6.428571428571428571428571428571 (decimal approximation)(*).

The recurring decimal 6.428571 can be written as a ratio of two integers having 45 as the numerator and 7 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.