What is 2.122323... as a Fraction?

2.1223 equals 210119900 as a fraction.

How do you turn 2.1223 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.1223 repeating into a fraction, begin writing this simple equation:

n = 2.1223 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (23) and 2 digits in the non-repeating part (12). Multiply both sides by 10⁴ = 10000.

10000 × n = 21223.23 (equation 2)

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

10000 × n = 21223.23
  100 × n = 2.1223
 9900 × n = 21011

n = 210119900 (answer)

2.1223 = 210119900 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 210119900 is also equal to 212119900 when expressed as a mixed number.

The repeating decimal 2.1223 (vinculum notation) has a repeated block length of 2. It is also represented as 2.12232323... (ellipsis notation) which equals approximately 2.122323232323 (decimal approximation)(*).

The recurring decimal 2.1223 can be written as a ratio of two integers having 21011 as the numerator and 9900 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.