What is 21.0142142142... as a Fraction?

21.142 equals 1049664995 as a fraction.

How do you turn 21.142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 21.142 repeating into a fraction, begin writing this simple equation:

n = 21.142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142), so multiply both sides by 10³ = 1000.

1000 × n = 21142.142 (equation 2)

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

1000 × n = 21142.142
   1 × n = 21.142
 999 × n = 21121

n = 1049664995 (answer)

21.142 = 1049664995 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 1049664995 is also equal to 21714995 when expressed as a mixed number.

The repeating decimal 21.142 (vinculum notation) has a repeated block length of 3. It is also represented as 21.142142142... (ellipsis notation) which equals approximately 21.142142142142142 (decimal approximation)(*).

The recurring decimal 21.142 can be written as a ratio of two integers having 104966 as the numerator and 4995 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.