What is 2.25714257142571425714257142... as a Fraction?

2.257142 equals 22571299999 as a fraction.

How do you turn 2.257142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.257142 repeating into a fraction, begin writing this simple equation:

n = 2.257142 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (57142) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 2257142.57142 (equation 2)

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

1000000 × n = 2257142.57142
     10 × n = 2.257142
 999990 × n = 2257120

2257120999990 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 10 (the GCF - greatest common factor).

n = 2257120999990 = 2257120 ÷ 10999990 ÷ 10 = 22571299999. So,

2.257142 = 22571299999 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 22571299999 is also equal to 22571499999 when expressed as a mixed number.

The repeating decimal 2.257142 (vinculum notation) has a repeated block length of 5. It is also represented as 2.2571425714257142... (ellipsis notation) which equals approximately 2.25714257142571425714257142 (decimal approximation)(*).

The recurring decimal 2.257142 can be written as a ratio of two integers having 225712 as the numerator and 99999 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.