What is 2.18428571428571428571428571428571428571... as a Fraction?

2.18428571 equals 1529700 as a fraction.

How do you turn 2.18428571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.18428571 repeating into a fraction, begin writing this simple equation:

n = 2.18428571 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (428571) and 2 digits in the non-repeating part (18). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 218428571.428571 (equation 2)

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

100000000 × n = 218428571.428571
      100 × n = 2.18428571
 99999900 × n = 218428353

21842835399999900 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 = 21842835399999900 = 218428353 ÷ 14285799999900 ÷ 142857 = 1529700. So,

2.18428571 = 1529700 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 1529700 is also equal to 2129700 when expressed as a mixed number.

The repeating decimal 2.18428571 (vinculum notation) has a repeated block length of 6. It is also represented as 2.18428571428571428571... (ellipsis notation) which equals approximately 2.18428571428571428571428571428571 (decimal approximation)(*).

The recurring decimal 2.18428571 can be written as a ratio of two integers having 1529 as the numerator and 700 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.