Converting Proper Fraction into Infinite Periodic Decimal

Suppose we have decimal fraction 2.73. If we add any number of zeros to the right, then value of fraction will not change: 2.73=2.730=2.7300=2.7300...0. This means that we can write the fraction 2.73 as decimal fraction with infinite set zeros i.e. 2.73=2.73000... . Here we have infinitely many decimal digits after dot. Such decimal fraction is called infinite decimal fraction.

Fact. Any proper fraction can be represented in the form of infinite decimal fraction.

For example, let′s take `3/14` and will divide the numerator by denominator, gradually we obtain decimal digits.

Thus, `3/14=0.214285714... ` .

Write out remains sequentially, which we obtain after division:2, 6, 4, 12, 8, 10, 12, 6... . All these remains are less than denominator, i.e. less than 14. This means that at some step of division must inevitably appear such remainder again, which was at the first step. So at the seventh step was remainder 2, which was at the first stet also. It is clear that as soon as appears remainder that already appeared, other remains will follow it in the same order as earlier.

Recurrent group of remains will lead to recurrent grupe of digits in the decimal record of number:

`3/14=0.2142857142857142857...` .

Recurrent group of digits (minimal) in the record of infinite decimal fraction after dot is called period and infinite decimal fraction, which has such period in one′s record is called periodic.

For brevity, we write the period once and take it into the brackets: `0.2142857142857142857...=0.2(142857)`.

When period begins at once after dot then such fraction is called purely periodic; when there are another decimal digits between the dot and period then such fraction is called mixed periodic.

So, `2.(23)=2.2323232323...` - purely periodic fraction; `2.73=2.73666...=2.73(6)` -mixed periodic fraction.