What is 0.86666... as a Fraction?

0.866 equals 1315 as a fraction.

How do you turn 0.866 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.866 repeating into a fraction, begin writing this simple equation:

n = 0.866 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (6) and 2 digits in the non-repeating part (86). Multiply both sides by 10³ = 1000.

1000 × n = 866.6 (equation 2)

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

1000 × n = 866.6
 100 × n = 0.866
 900 × n = 780

780900 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 60 (the GCF - greatest common factor).

n = 780900 = 780 ÷ 60900 ÷ 60 = 1315. So,

0.866 = 1315 as the lowest possible fraction.

The repeating decimal 0.866 (vinculum notation) has a repeated block length of 1. It is also represented as 0.86666... (ellipsis notation) which equals approximately 0.8666666 (decimal approximation)(*).

The recurring decimal 0.866 can be written as a ratio of two integers having 13 as the numerator and 15 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.