What is 1.41428142814281428... as a Fraction?

1.41428 equals 2356916665 as a fraction.

How do you turn 1.41428 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.41428 repeating into a fraction, begin writing this simple equation:

n = 1.41428 (equation 1)

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

100000 × n = 141428.1428 (equation 2)

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

100000 × n = 141428.1428
    10 × n = 1.41428
 99990 × n = 141414

14141499990 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 6 (the GCF - greatest common factor).

n = 14141499990 = 141414 ÷ 699990 ÷ 6 = 2356916665. So,

1.41428 = 2356916665 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 2356916665 is also equal to 1690416665 when expressed as a mixed number.

The repeating decimal 1.41428 (vinculum notation) has a repeated block length of 4. It is also represented as 1.4142814281428... (ellipsis notation) which equals approximately 1.414281428142814281428 (decimal approximation)(*).

The recurring decimal 1.41428 can be written as a ratio of two integers having 23569 as the numerator and 16665 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.