What is 2.110222... as a Fraction?

2.1102 equals 23741125 as a fraction.

How do you turn 2.1102 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.1102 repeating into a fraction, begin writing this simple equation:

n = 2.1102 (equation 1)

Step 2: Notice that there is 1 digit in the repeating block (2) and 3 digits in the non-repeating part (110). Multiply both sides by 10⁴ = 10000.

10000 × n = 21102.2 (equation 2)

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

10000 × n = 21102.2
 1000 × n = 2.1102
 9000 × n = 18992

189929000 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 8 (the GCF - greatest common factor).

n = 189929000 = 18992 ÷ 89000 ÷ 8 = 23741125. So,

2.1102 = 23741125 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 23741125 is also equal to 21241125 when expressed as a mixed number.

The repeating decimal 2.1102 (vinculum notation) has a repeated block length of 1. It is also represented as 2.110222... (ellipsis notation) which equals approximately 2.11022222 (decimal approximation)(*).

The recurring decimal 2.1102 can be written as a ratio of two integers having 2374 as the numerator and 1125 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.