What is 1.45714571457145714... as a Fraction?

1.45714 equals 145709999 as a fraction.

How do you turn 1.45714 repeating into a fraction?

Detailed Answer:

Step 1: To convert 1.45714 repeating into a fraction, begin writing this simple equation:

n = 1.45714 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (5714) and 1 digit in the non-repeating part (4). Multiply both sides by 10⁵ = 100000.

100000 × n = 145714.5714 (equation 2)

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

100000 × n = 145714.5714
    10 × n = 1.45714
 99990 × n = 145700

14570099990 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 10 (the GCF - greatest common factor).

n = 14570099990 = 145700 ÷ 1099990 ÷ 10 = 145709999. So,

1.45714 = 145709999 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 145709999 is also equal to 145719999 when expressed as a mixed number.

The repeating decimal 1.45714 (vinculum notation) has a repeated block length of 4. It is also represented as 1.4571457145714... (ellipsis notation) which equals approximately 1.457145714571457145714 (decimal approximation)(*).

The recurring decimal 1.45714 can be written as a ratio of two integers having 14570 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.