What is 0.227142714271427142... as a Fraction?

0.227142 equals 1135649995 as a fraction.

How do you turn 0.227142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.227142 repeating into a fraction, begin writing this simple equation:

n = 0.227142 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (7142) and 2 digits in the non-repeating part (22). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 227142.7142 (equation 2)

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

1000000 × n = 227142.7142
    100 × n = 0.227142
 999900 × n = 227120

227120999900 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 20 (the GCF - greatest common factor).

n = 227120999900 = 227120 ÷ 20999900 ÷ 20 = 1135649995. So,

0.227142 = 1135649995 as the lowest possible fraction.

The repeating decimal 0.227142 (vinculum notation) has a repeated block length of 4. It is also represented as 0.22714271427142... (ellipsis notation) which equals approximately 0.2271427142714271427142 (decimal approximation)(*).

The recurring decimal 0.227142 can be written as a ratio of two integers having 11356 as the numerator and 49995 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.