What is 13.227142714271427142... as a Fraction?

13.227142 equals 66129149995 as a fraction.

How do you turn 13.227142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.227142 repeating into a fraction, begin writing this simple equation:

n = 13.227142 (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 = 13227142.7142 (equation 2)

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

1000000 × n = 13227142.7142
    100 × n = 13.227142
 999900 × n = 13225820

13225820999900 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 = 13225820999900 = 13225820 ÷ 20999900 ÷ 20 = 66129149995. So,

13.227142 = 66129149995 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 66129149995 is also equal to 131135649995 when expressed as a mixed number.

The repeating decimal 13.227142 (vinculum notation) has a repeated block length of 4. It is also represented as 13.22714271427142... (ellipsis notation) which equals approximately 13.2271427142714271427142 (decimal approximation)(*).

The recurring decimal 13.227142 can be written as a ratio of two integers having 661291 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.