What is 0.145714571457145714... as a Fraction?

0.145714 equals 14579999 as a fraction.

How do you turn 0.145714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 0.145714 repeating into a fraction, begin writing this simple equation:

n = 0.145714 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (5714) and 2 digits in the non-repeating part (14). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 145714.5714 (equation 2)

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

1000000 × n = 145714.5714
    100 × n = 0.145714
 999900 × n = 145700

145700999900 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 100 (the GCF - greatest common factor).

n = 145700999900 = 145700 ÷ 100999900 ÷ 100 = 14579999. So,

0.145714 = 14579999 as the lowest possible fraction.

The repeating decimal 0.145714 (vinculum notation) has a repeated block length of 4. It is also represented as 0.14571457145714... (ellipsis notation) which equals approximately 0.1457145714571457145714 (decimal approximation)(*).

The recurring decimal 0.145714 can be written as a ratio of two integers having 1457 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.