What is 0.8076923076923076923076923076923076923... as a Fraction?

0.8076923 equals 2126 as a fraction.

How do you turn 0.8076923 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.8076923 repeating into a fraction, begin writing this simple equation:

n = 0.8076923 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (076923) and 1 digit in the non-repeating part (8). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 8076923.076923 (equation 2)

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

10000000 × n = 8076923.076923
      10 × n = 0.8076923
 9999990 × n = 8076915

80769159999990 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 384615 (the GCF - greatest common factor).

n = 80769159999990 = 8076915 ÷ 3846159999990 ÷ 384615 = 2126. So,

0.8076923 = 2126 as the lowest possible fraction.

The repeating decimal 0.8076923 (vinculum notation) has a repeated block length of 6. It is also represented as 0.8076923076923076923... (ellipsis notation) which equals approximately 0.8076923076923076923076923076923 (decimal approximation)(*).

The recurring decimal 0.8076923 can be written as a ratio of two integers having 21 as the numerator and 26 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.