What is 3.14384384384... as a Fraction?

3.14384 equals 104693330 as a fraction.

How do you turn 3.14384 repeating into a fraction?

Detailed Answer:

Step 1: To convert 3.14384 repeating into a fraction, begin writing this simple equation:

n = 3.14384 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (384) and 2 digits in the non-repeating part (14). Multiply both sides by 10⁵ = 100000.

100000 × n = 314384.384 (equation 2)

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

100000 × n = 314384.384
   100 × n = 3.14384
 99900 × n = 314070

31407099900 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 30 (the GCF - greatest common factor).

n = 31407099900 = 314070 ÷ 3099900 ÷ 30 = 104693330. So,

3.14384 = 104693330 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 104693330 is also equal to 34793330 when expressed as a mixed number.

The repeating decimal 3.14384 (vinculum notation) has a repeated block length of 3. It is also represented as 3.14384384384... (ellipsis notation) which equals approximately 3.14384384384384384 (decimal approximation)(*).

The recurring decimal 3.14384 can be written as a ratio of two integers having 10469 as the numerator and 3330 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.