# Which is 1 9999 as a fraction

If you want to convert given periodic decimal numbers into fractions, you first have to consider the reciprocal values ​​of \ (9 \), \ (99 \), \ (999 \), \ (9999 \) etc. It is good practice to calculate these numbers once by written division. We have:

\ begin {align *}
\ frac {1} {9} & = 0.11111 ... && = 0, \ overline {1} \
\ frac {1} {99} & = 0.010101 ... && = 0, \ overline {01} \
\ frac {1} {999} & = 0.001001001 ... && = 0, \ overline {001}
\ end {align *}

These decimal numbers are, so to speak, the prototypes of periodic decimal numbers: if one has, for example, \ (0, \ overline {301} = 0.301301301301 ... \), the number can also be written as
\ begin {align *}
0.301301301301… & = 301 \ cdot 0.001001001001 \
& = 301 \ cdot \ frac {1} {999} \ & = \ frac {301} {999}.
\ end {align *}
\ (0, \ overline {5} = 0.5555555… \) can be written as
\ begin {align *} 0.55555 ... & = 5 \ cdot 0.111111 ... \ & = 5 \ cdot \ frac {1} {9} \ & = \ frac {5} {9} . \ end {align *}

Numbers whose period begins directly after the decimal point are called purely periodically. If you want to convert periodic numbers into fractions whose period does not start immediately, such as \ (0.41 \ overline {6} = 0.4166666 ... \), you proceed as follows: First you divide the number into a terminating and a periodic part: \ begin {align *} 0.416 = 0.41 + 0.00 \ overline {6}. \ end {align *}
As you learned in previous chapters, \ (0,41 \) is the same as \ (\ frac {41} {100} \). With \ (0.00 \ overline {6} \) the product of a power of ten and a purely periodic decimal number is written, here: \ (0.00 \ overline {6} = 0.01 \ cdot 0, \ overline { 6}. \)
\ (0, \ overline {6} \) can be used as \ begin {align *} 0, \ overline {6} = 6 \ cdot 0, \ overline {1} = 6 \ cdot \ frac {1} according to the above consideration {9} = \ frac {6} {9} = understand \ frac {2} {3} \ end {align *}. So we have a total of:

\ begin {align *}
0.41 \ overline {6} & = 0.41 + 0.00 \ overline {6} \
& = \ frac {41} {100} +0.01 \ cdot 0, \ overline {6} \
& = \ frac {41} {100} + \ frac {1} {100} \ cdot \ frac {2} {3} \
& = \ frac {41} {100} + \ frac {2} {300} \
& = \ frac {123} {300} + \ frac {2} {300} \
& = \ frac {125} {300} \
& = \ frac {5} {12}
\ end {align *}

Another example: \ (2,5 \ overline {27} \) should be converted into a fraction, one calculates:
\ begin {align *} 2.5 \ overline {27} & = 2.5 + 0.0 \ overline {27} \
& = \ frac {25} {10} +0.0 \ overline {27} \
& = \ frac {5} {2} +0.1 \ cdot 0, \ overline {27} \
& = \ frac {5} {2} + \ frac {1} {10} \ cdot \ frac {27} {99} \
& = \ frac {5} {2} + \ frac {1} {10} \ cdot \ frac {3} {11} \
& = \ frac {5} {2} + \ frac {3} {110} \
& = \ frac {275} {110} + \ frac {3} {110} \
& = \ frac {278} {110} \
& = \ frac {139} {55}.
\ end {align *}