What is 2.3571428571428571428571428571428571428... as a Fraction?

2.3571428 equals 3314 as a fraction.

How do you turn 2.3571428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.3571428 repeating into a fraction, begin writing this simple equation:

n = 2.3571428 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (571428) and 1 digit in the non-repeating part (3). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 23571428.571428 (equation 2)

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

10000000 × n = 23571428.571428
      10 × n = 2.3571428
 9999990 × n = 23571405

235714059999990 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 714285 (the GCF - greatest common factor).

n = 235714059999990 = 23571405 ÷ 7142859999990 ÷ 714285 = 3314. So,

2.3571428 = 3314 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 3314 is also equal to 2514 when expressed as a mixed number.

The repeating decimal 2.3571428 (vinculum notation) has a repeated block length of 6. It is also represented as 2.3571428571428571428... (ellipsis notation) which equals approximately 2.3571428571428571428571428571428 (decimal approximation)(*).

The recurring decimal 2.3571428 can be written as a ratio of two integers having 33 as the numerator and 14 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.