What is 2.2145454545... as a Fraction?

2.2145 equals 609275 as a fraction.

How do you turn 2.2145 repeating into a fraction?

Detailed Answer:

Step 1: To convert 2.2145 repeating into a fraction, begin writing this simple equation:

n = 2.2145 (equation 1)

Step 2: Notice that there are 2 digits in the repeating block (45) and 2 digits in the non-repeating part (21). Multiply both sides by 10⁴ = 10000.

10000 × n = 22145.45 (equation 2)

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

10000 × n = 22145.45
  100 × n = 2.2145
 9900 × n = 21924

219249900 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 36 (the GCF - greatest common factor).

n = 219249900 = 21924 ÷ 369900 ÷ 36 = 609275. So,

2.2145 = 609275 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 609275 is also equal to 259275 when expressed as a mixed number.

The repeating decimal 2.2145 (vinculum notation) has a repeated block length of 2. It is also represented as 2.21454545... (ellipsis notation) which equals approximately 2.214545454545 (decimal approximation)(*).

The recurring decimal 2.2145 can be written as a ratio of two integers having 609 as the numerator and 275 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.