What is 4.137142871428714287142871428... as a Fraction?

4.1371428 equals 82742031999980 as a fraction.

How do you turn 4.1371428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 4.1371428 repeating into a fraction, begin writing this simple equation:

n = 4.1371428 (equation 1)

Step 2: Notice that there are 5 digits in the repeating block (71428) and 2 digits in the non-repeating part (13). Multiply both sides by 10⁷ = 10000000.

10000000 × n = 41371428.71428 (equation 2)

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

10000000 × n = 41371428.71428
     100 × n = 4.1371428
 9999900 × n = 41371015

413710159999900 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 5 (the GCF - greatest common factor).

n = 413710159999900 = 41371015 ÷ 59999900 ÷ 5 = 82742031999980. So,

4.1371428 = 82742031999980 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 82742031999980 is also equal to 42742831999980 when expressed as a mixed number.

The repeating decimal 4.1371428 (vinculum notation) has a repeated block length of 5. It is also represented as 4.13714287142871428... (ellipsis notation) which equals approximately 4.137142871428714287142871428 (decimal approximation)(*).

The recurring decimal 4.1371428 can be written as a ratio of two integers having 8274203 as the numerator and 1999980 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.