What is 0.21076923076923076923076923076923076923... as a Fraction?

0.21076923 equals 137650 as a fraction.

How do you turn 0.21076923 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.21076923 repeating into a fraction, begin writing this simple equation:

n = 0.21076923 (equation 1)

Step 2: Notice that there are 6 digits in the repeating block (076923) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁸ = 100000000.

100000000 × n = 21076923.076923 (equation 2)

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

100000000 × n = 21076923.076923
      100 × n = 0.21076923
 99999900 × n = 21076902

2107690299999900 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 153846 (the GCF - greatest common factor).

n = 2107690299999900 = 21076902 ÷ 15384699999900 ÷ 153846 = 137650. So,

0.21076923 = 137650 as the lowest possible fraction.

The repeating decimal 0.21076923 (vinculum notation) has a repeated block length of 6. It is also represented as 0.21076923076923076923... (ellipsis notation) which equals approximately 0.21076923076923076923076923076923 (decimal approximation)(*).

The recurring decimal 0.21076923 can be written as a ratio of two integers having 137 as the numerator and 650 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.