What is 20.4461461461... as a Fraction?

20.4461 equals 2042579990 as a fraction.

How do you turn 20.4461 repeating into a fraction?

Detailed Answer:

Step 1: To convert 20.4461 repeating into a fraction, begin writing this simple equation:

n = 20.4461 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (461) and 1 digit in the non-repeating part (4). Multiply both sides by 10⁴ = 10000.

10000 × n = 204461.461 (equation 2)

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

10000 × n = 204461.461
   10 × n = 20.4461
 9990 × n = 204257

n = 2042579990 (answer)

20.4461 = 2042579990 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 2042579990 is also equal to 2044579990 when expressed as a mixed number.

The repeating decimal 20.4461 (vinculum notation) has a repeated block length of 3. It is also represented as 20.4461461461... (ellipsis notation) which equals approximately 20.4461461461461461 (decimal approximation)(*).

The recurring decimal 20.4461 can be written as a ratio of two integers having 204257 as the numerator and 9990 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.