What is 2.181428514285142851428514285... as a Fraction?

2.1814285 equals 218140679999900 as a fraction.

How do you turn 2.1814285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.1814285 repeating into a fraction, begin writing this simple equation:

n = 2.1814285 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (14285) and 2 digits in the non-repeating part (18). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 21814285.14285 (equation 2)

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

10000000 × n = 21814285.14285
     100 × n = 2.1814285
 9999900 × n = 21814067

n = 218140679999900 (answer)

2.1814285 = 218140679999900 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 218140679999900 is also equal to 218142679999900 when expressed as a mixed number.

The repeating decimal 2.1814285 (vinculum notation) has a repeated block length of 5. It is also represented as 2.18142851428514285... (ellipsis notation) which equals approximately 2.181428514285142851428514285 (decimal approximation)(*).

The recurring decimal 2.1814285 can be written as a ratio of two integers having 21814067 as the numerator and 9999900 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.