What is 8.107142714271427142... as a Fraction?

8.107142 equals 2026583249975 as a fraction.

How do you turn 8.107142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 8.107142 repeating into a fraction, begin writing this simple equation:

n = 8.107142 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (7142) and 2 digits in the non-repeating part (10). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 8107142.7142 (equation 2)

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

1000000 × n = 8107142.7142
    100 × n = 8.107142
 999900 × n = 8106332

8106332999900 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 4 (the GCF - greatest common factor).

n = 8106332999900 = 8106332 ÷ 4999900 ÷ 4 = 2026583249975. So,

8.107142 = 2026583249975 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 2026583249975 is also equal to 826783249975 when expressed as a mixed number.

The repeating decimal 8.107142 (vinculum notation) has a repeated block length of 4. It is also represented as 8.10714271427142... (ellipsis notation) which equals approximately 8.1071427142714271427142 (decimal approximation)(*).

The recurring decimal 8.107142 can be written as a ratio of two integers having 2026583 as the numerator and 249975 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.