What is 2.220606... as a Fraction?

2.2206 equals 1832825 as a fraction.

How do you turn 2.2206 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.2206 repeating into a fraction, begin writing this simple equation:

n = 2.2206 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (06) and 2 digits in the non-repeating part (22). Multiply both sides by 10⁴ = 10000.

10000 × n = 22206.06 (equation 2)

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

10000 × n = 22206.06
  100 × n = 2.2206
 9900 × n = 21984

219849900 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 12 (the GCF - greatest common factor).

n = 219849900 = 21984 ÷ 129900 ÷ 12 = 1832825. So,

2.2206 = 1832825 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 1832825 is also equal to 2182825 when expressed as a mixed number.

The repeating decimal 2.2206 (vinculum notation) has a repeated block length of 2. It is also represented as 2.22060606... (ellipsis notation) which equals approximately 2.220606060606 (decimal approximation)(*).

The recurring decimal 2.2206 can be written as a ratio of two integers having 1832 as the numerator and 825 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.