What is 0.45714257142571425714257142... as a Fraction?

0.457142 equals 228569499995 as a fraction.

How do you turn 0.457142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.457142 repeating into a fraction, begin writing this simple equation:

n = 0.457142 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (57142) and 1 digit in the non-repeating part (4). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 457142.57142 (equation 2)

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

1000000 × n = 457142.57142
     10 × n = 0.457142
 999990 × n = 457138

457138999990 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 2 (the GCF - greatest common factor).

n = 457138999990 = 457138 ÷ 2999990 ÷ 2 = 228569499995. So,

0.457142 = 228569499995 as the lowest possible fraction.

The repeating decimal 0.457142 (vinculum notation) has a repeated block length of 5. It is also represented as 0.4571425714257142... (ellipsis notation) which equals approximately 0.45714257142571425714257142 (decimal approximation)(*).

The recurring decimal 0.457142 can be written as a ratio of two integers having 228569 as the numerator and 499995 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.