What is 2.2428571428571428571428571428571428571... as a Fraction?

2.2428571 equals 15770 as a fraction.

How do you turn 2.2428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.2428571 repeating into a fraction, begin writing this simple equation:

n = 2.2428571 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (428571) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 22428571.428571 (equation 2)

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

10000000 × n = 22428571.428571
      10 × n = 2.2428571
 9999990 × n = 22428549

224285499999990 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 = 224285499999990 = 22428549 ÷ 1428579999990 ÷ 142857 = 15770. So,

2.2428571 = 15770 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 15770 is also equal to 21770 when expressed as a mixed number.

The repeating decimal 2.2428571 (vinculum notation) has a repeated block length of 6. It is also represented as 2.2428571428571428571... (ellipsis notation) which equals approximately 2.2428571428571428571428571428571 (decimal approximation)(*).

The recurring decimal 2.2428571 can be written as a ratio of two integers having 157 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.