What is 2.12384384384... as a Fraction?

2.12384 equals 176818325 as a fraction.

How do you turn 2.12384 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.12384 repeating into a fraction, begin writing this simple equation:

n = 2.12384 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (384) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁵ = 100000.

100000 × n = 212384.384 (equation 2)

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

100000 × n = 212384.384
   100 × n = 2.12384
 99900 × n = 212172

21217299900 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 12 (the GCF - greatest common factor).

n = 21217299900 = 212172 ÷ 1299900 ÷ 12 = 176818325. So,

2.12384 = 176818325 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 176818325 is also equal to 210318325 when expressed as a mixed number.

The repeating decimal 2.12384 (vinculum notation) has a repeated block length of 3. It is also represented as 2.12384384384... (ellipsis notation) which equals approximately 2.12384384384384384 (decimal approximation)(*).

The recurring decimal 2.12384 can be written as a ratio of two integers having 17681 as the numerator and 8325 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.