What is 2.31428142814281428... as a Fraction?

2.31428 equals 154276666 as a fraction.

How do you turn 2.31428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.31428 repeating into a fraction, begin writing this simple equation:

n = 2.31428 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (1428) and 1 digit in the non-repeating part (3). Multiply both sides by 10⁵ = 100000.

100000 × n = 231428.1428 (equation 2)

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

100000 × n = 231428.1428
    10 × n = 2.31428
 99990 × n = 231405

23140599990 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 15 (the GCF - greatest common factor).

n = 23140599990 = 231405 ÷ 1599990 ÷ 15 = 154276666. So,

2.31428 = 154276666 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 154276666 is also equal to 220956666 when expressed as a mixed number.

The repeating decimal 2.31428 (vinculum notation) has a repeated block length of 4. It is also represented as 2.3142814281428... (ellipsis notation) which equals approximately 2.314281428142814281428 (decimal approximation)(*).

The recurring decimal 2.31428 can be written as a ratio of two integers having 15427 as the numerator and 6666 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.