Periodic fraction

not all ordinary fractions can be represented as a final decimal fraction.

For example, if we divide 2 by 3, we first get zero integers, then six tenths, and then when we divide, the remainder 2 will be repeated all the time, and in private - the number 6.

Such a division cannot be completed without remainder, and therefore a 2/3 fraction cannot be represented as a final decimal fraction.

In a brief record of a periodic fraction, a repeating number (or group of numbers) is written in brackets. This number (or group of numbers) is called the fraction period .

Instead of 0.666 ... they write 0, (6) and read "zero zero and six in the period".

The translation of the periodic fraction in the ordinary

Periodic infinite decimal fraction can be converted to ordinary fraction .

Consider a periodic fraction of 10.0219 (37).

• We count the number of digits in the period of the decimal fraction. We denote the number of digits for the letter k. We have k = 2.
• We count the number of digits after the decimal point, but before the period of the decimal fraction. We denote the number of digits for the letter m. We have m = 4.
• We write down all figures after a comma ( including figures from the period ) in the form of a natural number. If at the beginning, before the first significant digit, go zeros, then we discard them. We denote the resulting number by the letter a.
  a = 021937 = 21 937
  
• Now we write down all the figures after the comma, but before the period, in the form of a natural number. If zeros first go to the first significant digit, then we discard them. Denote the resulting number by the letter b.
  b = 0219 = 219
  
• Substitute the values ​​found in the formula, where Y is the integer part of the infinite periodic fraction . We have Y = 10.

An example of the conversion of a periodic fraction to an ordinary

So, we substitute all found values ​​in the formula above and we receive an ordinary fraction. The received answer can always be checked on a regular calculator.