What is 21.1101010101... as a Fraction?

21.1101 equals 20899990 as a fraction.

How do you turn 21.1101 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.1101 repeating into a fraction, begin writing this simple equation:

n = 21.1101 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (01) and 2 digits in the non-repeating part (11). Multiply both sides by 10⁴ = 10000.

10000 × n = 211101.01 (equation 2)

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

10000 × n = 211101.01
  100 × n = 21.1101
 9900 × n = 208990

2089909900 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 10 (the GCF - greatest common factor).

n = 2089909900 = 208990 ÷ 109900 ÷ 10 = 20899990. So,

21.1101 = 20899990 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 20899990 is also equal to 21109990 when expressed as a mixed number.

The repeating decimal 21.1101 (vinculum notation) has a repeated block length of 2. It is also represented as 21.11010101... (ellipsis notation) which equals approximately 21.110101010101 (decimal approximation)(*).

The recurring decimal 21.1101 can be written as a ratio of two integers having 20899 as the numerator and 990 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.