What is 21.23846384638463846... as a Fraction?

21.23846 equals 35393916665 as a fraction.

How do you turn 21.23846 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.23846 repeating into a fraction, begin writing this simple equation:

n = 21.23846 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (3846) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁵ = 100000.

100000 × n = 2123846.3846 (equation 2)

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

100000 × n = 2123846.3846
    10 × n = 21.23846
 99990 × n = 2123634

212363499990 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 = 212363499990 = 2123634 ÷ 699990 ÷ 6 = 35393916665. So,

21.23846 = 35393916665 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 35393916665 is also equal to 21397416665 when expressed as a mixed number.

The repeating decimal 21.23846 (vinculum notation) has a repeated block length of 4. It is also represented as 21.2384638463846... (ellipsis notation) which equals approximately 21.238463846384638463846 (decimal approximation)(*).

The recurring decimal 21.23846 can be written as a ratio of two integers having 353939 as the numerator and 16665 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.