What is 2.11428142814281428... as a Fraction?

2.11428 equals 7046933330 as a fraction.

How do you turn 2.11428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.11428 repeating into a fraction, begin writing this simple equation:

n = 2.11428 (equation 1)

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

100000 × n = 211428.1428 (equation 2)

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

100000 × n = 211428.1428
    10 × n = 2.11428
 99990 × n = 211407

21140799990 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 3 (the GCF - greatest common factor).

n = 21140799990 = 211407 ÷ 399990 ÷ 3 = 7046933330. So,

2.11428 = 7046933330 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 7046933330 is also equal to 2380933330 when expressed as a mixed number.

The repeating decimal 2.11428 (vinculum notation) has a repeated block length of 4. It is also represented as 2.1142814281428... (ellipsis notation) which equals approximately 2.114281428142814281428 (decimal approximation)(*).

The recurring decimal 2.11428 can be written as a ratio of two integers having 70469 as the numerator and 33330 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.