What is 12.123846384638463846... as a Fraction?

12.123846 equals 2020439166650 as a fraction.

How do you turn 12.123846 repeating into a fraction?

Detailed Answer:

Step 1: To convert 12.123846 repeating into a fraction, begin writing this simple equation:

n = 12.123846 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (3846) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 12123846.3846 (equation 2)

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

1000000 × n = 12123846.3846
    100 × n = 12.123846
 999900 × n = 12122634

12122634999900 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 6 (the GCF - greatest common factor).

n = 12122634999900 = 12122634 ÷ 6999900 ÷ 6 = 2020439166650. So,

12.123846 = 2020439166650 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 2020439166650 is also equal to 1220639166650 when expressed as a mixed number.

The repeating decimal 12.123846 (vinculum notation) has a repeated block length of 4. It is also represented as 12.12384638463846... (ellipsis notation) which equals approximately 12.1238463846384638463846 (decimal approximation)(*).

The recurring decimal 12.123846 can be written as a ratio of two integers having 2020439 as the numerator and 166650 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.