What is 2.212857128571285712857128571... as a Fraction?

2.2128571 equals 442567199998 as a fraction.

How do you turn 2.2128571 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.2128571 repeating into a fraction, begin writing this simple equation:

n = 2.2128571 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (28571) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 22128571.28571 (equation 2)

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

10000000 × n = 22128571.28571
     100 × n = 2.2128571
 9999900 × n = 22128350

221283509999900 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 50 (the GCF - greatest common factor).

n = 221283509999900 = 22128350 ÷ 509999900 ÷ 50 = 442567199998. So,

2.2128571 = 442567199998 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 442567199998 is also equal to 242571199998 when expressed as a mixed number.

The repeating decimal 2.2128571 (vinculum notation) has a repeated block length of 5. It is also represented as 2.21285712857128571... (ellipsis notation) which equals approximately 2.212857128571285712857128571 (decimal approximation)(*).

The recurring decimal 2.2128571 can be written as a ratio of two integers having 442567 as the numerator and 199998 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.