What is 2.12285714285714285714285714285714285714... as a Fraction?

2.12285714 equals 743350 as a fraction.

How do you turn 2.12285714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.12285714 repeating into a fraction, begin writing this simple equation:

n = 2.12285714 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (285714) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 212285714.285714 (equation 2)

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

100000000 × n = 212285714.285714
      100 × n = 2.12285714
 99999900 × n = 212285502

21228550299999900 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 285714 (the GCF - greatest common factor).

n = 21228550299999900 = 212285502 ÷ 28571499999900 ÷ 285714 = 743350. So,

2.12285714 = 743350 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 743350 is also equal to 243350 when expressed as a mixed number.

The repeating decimal 2.12285714 (vinculum notation) has a repeated block length of 6. It is also represented as 2.12285714285714285714... (ellipsis notation) which equals approximately 2.12285714285714285714285714285714 (decimal approximation)(*).

The recurring decimal 2.12285714 can be written as a ratio of two integers having 743 as the numerator and 350 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.