What is 2.142307230723072307... as a Fraction?

2.142307 equals 714031333300 as a fraction.

How do you turn 2.142307 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.142307 repeating into a fraction, begin writing this simple equation:

n = 2.142307 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (2307) and 2 digits in the non-repeating part (14). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 2142307.2307 (equation 2)

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

1000000 × n = 2142307.2307
    100 × n = 2.142307
 999900 × n = 2142093

2142093999900 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 3 (the GCF - greatest common factor).

n = 2142093999900 = 2142093 ÷ 3999900 ÷ 3 = 714031333300. So,

2.142307 = 714031333300 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 714031333300 is also equal to 247431333300 when expressed as a mixed number.

The repeating decimal 2.142307 (vinculum notation) has a repeated block length of 4. It is also represented as 2.14230723072307... (ellipsis notation) which equals approximately 2.1423072307230723072307 (decimal approximation)(*).

The recurring decimal 2.142307 can be written as a ratio of two integers having 714031 as the numerator and 333300 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.