What is 0.4142142142... as a Fraction?

0.4142 equals 20694995 as a fraction.

How do you turn 0.4142 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.4142 repeating into a fraction, begin writing this simple equation:

n = 0.4142 (equation 1)

Step 2: Notice that there are 3 digits in the repeating block (142) and 1 digit in the non-repeating part (4). Multiply both sides by 10⁴ = 10000.

10000 × n = 4142.142 (equation 2)

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

10000 × n = 4142.142
   10 × n = 0.4142
 9990 × n = 4138

41389990 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 2 (the GCF - greatest common factor).

n = 41389990 = 4138 ÷ 29990 ÷ 2 = 20694995. So,

0.4142 = 20694995 as the lowest possible fraction.

The repeating decimal 0.4142 (vinculum notation) has a repeated block length of 3. It is also represented as 0.4142142142... (ellipsis notation) which equals approximately 0.4142142142142142 (decimal approximation)(*).

The recurring decimal 0.4142 can be written as a ratio of two integers having 2069 as the numerator and 4995 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.