What is 4.5714571457145714... as a Fraction?

4.5714 equals 457109999 as a fraction.

How do you turn 4.5714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 4.5714 repeating into a fraction, begin writing this simple equation:

n = 4.5714 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (5714), so multiply both sides by 10⁴ = 10000.

10000 × n = 45714.5714 (equation 2)

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

10000 × n = 45714.5714
    1 × n = 4.5714
 9999 × n = 45710

n = 457109999 (answer)

4.5714 = 457109999 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 457109999 is also equal to 457149999 when expressed as a mixed number.

The repeating decimal 4.5714 (vinculum notation) has a repeated block length of 4. It is also represented as 4.571457145714... (ellipsis notation) which equals approximately 4.57145714571457145714 (decimal approximation)(*).

The recurring decimal 4.5714 can be written as a ratio of two integers having 45710 as the numerator and 9999 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.