What is 2.104285428542854285... as a Fraction?

2.104285 equals 8416339996 as a fraction.

How do you turn 2.104285 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.104285 repeating into a fraction, begin writing this simple equation:

n = 2.104285 (equation 1)

Step 2: Notice that there are 4 digits in the repeating block (4285) and 2 digits in the non-repeating part (10). Multiply both sides by 10⁶ = 1000000.

1000000 × n = 2104285.4285 (equation 2)

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

1000000 × n = 2104285.4285
    100 × n = 2.104285
 999900 × n = 2104075

2104075999900 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 25 (the GCF - greatest common factor).

n = 2104075999900 = 2104075 ÷ 25999900 ÷ 25 = 8416339996. So,

2.104285 = 8416339996 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 8416339996 is also equal to 2417139996 when expressed as a mixed number.

The repeating decimal 2.104285 (vinculum notation) has a repeated block length of 4. It is also represented as 2.10428542854285... (ellipsis notation) which equals approximately 2.1042854285428542854285 (decimal approximation)(*).

The recurring decimal 2.104285 can be written as a ratio of two integers having 84163 as the numerator and 39996 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.