What is 2.163846384638463846... as a Fraction?

2.163846 equals 7212133330 as a fraction.

How do you turn 2.163846 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.163846 repeating into a fraction, begin writing this simple equation:

n = 2.163846 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (3846) and 2 digits in the non-repeating part (16). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 2163846.3846 (equation 2)

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

1000000 × n = 2163846.3846
    100 × n = 2.163846
 999900 × n = 2163630

2163630999900 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 = 2163630999900 = 2163630 ÷ 30999900 ÷ 30 = 7212133330. So,

2.163846 = 7212133330 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 7212133330 is also equal to 2546133330 when expressed as a mixed number.

The repeating decimal 2.163846 (vinculum notation) has a repeated block length of 4. It is also represented as 2.16384638463846... (ellipsis notation) which equals approximately 2.1638463846384638463846 (decimal approximation)(*).

The recurring decimal 2.163846 can be written as a ratio of two integers having 72121 as the numerator and 33330 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.