What is 13.22307230723072307... as a Fraction?

13.22307 equals 881456666 as a fraction.

How do you turn 13.22307 repeating into a fraction?

Detailed Answer:

Step 1: To convert 13.22307 repeating into a fraction, begin writing this simple equation:

n = 13.22307 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (2307) and 1 digit in the non-repeating part (2). Multiply both sides by 10⁵ = 100000.

100000 × n = 1322307.2307 (equation 2)

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

100000 × n = 1322307.2307
    10 × n = 13.22307
 99990 × n = 1322175

132217599990 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 15 (the GCF - greatest common factor).

n = 132217599990 = 1322175 ÷ 1599990 ÷ 15 = 881456666. So,

13.22307 = 881456666 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 881456666 is also equal to 1314876666 when expressed as a mixed number.

The repeating decimal 13.22307 (vinculum notation) has a repeated block length of 4. It is also represented as 13.2230723072307... (ellipsis notation) which equals approximately 13.223072307230723072307 (decimal approximation)(*).

The recurring decimal 13.22307 can be written as a ratio of two integers having 88145 as the numerator and 6666 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.