How come 0 9 reps equals 1?

Periodic decimal numbers

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In these explanations you will learn what properties periodic decimal numbers have and how you can convert them into fractions.

What is a periodic decimal?

You can recognize a periodic decimal number by the fact that a digit or a sequence of digits is repeated after the decimal point. The repetitions can start at the first decimal place, but they can also start later.

How is a periodic decimal number created?

Periodic decimal numbers are created when dividing by the remainder when you divide the remainder further. If the divisor only has the prime factors 2 or 5, you get a decimal number with a finite number of decimal places. If the divisior is 3 or 7, 11, ..., you get a periodic decimal number.

Rounding a periodic decimal number

In order to be able to calculate with periodic decimal numbers, it is helpful to round them to a suitable decimal place. So that you know whether you have to round up or down, you have to determine which number is to the right of the rounding point.
Round to tenth.
Round to hundredths.
Round to the thousandth.
Round to hundredths.

Compare periodic decimal numbers

If you want to compare periodic decimal numbers with one another or with other decimal numbers, you compare their place values ​​with one another. You start from the left. The first different significance must be compared. If this first place value is not in the period, then you proceed in the same way as when comparing non-periodic decimal numbers.
Compare and 0.25.
   
Compare and 7.51.
  <  
If this first place value is in the period, then you dissolve the period, that is, you write the periodic decimal number without a period line.
Compare and.
   
Compare and.
   

Pure period and mixed period decimal numbers

Purely periodic decimal numbers are decimal numbers where the period begins directly after the decimal point.
0,294294294... =
Mixed periodic decimal numbers are decimal numbers with at least one digit between the period and the comma.
  ... =

Convert pure period decimal numbers to fractions

Pure period and mixed period decimal numbers are treated differently when converting to fractions. When converting purely periodic decimal numbers, the denominator of the fraction you are looking for is always 9, 99, 999, ... The length of the period shows you the number of nines in the denominator. The number is written under the period line in the numerator of the fraction you are looking for. If there is a natural number in front of the comma, you get a mixed number with this natural number as a whole.
Convert to a fraction.
Convert to a fraction.

Convert mixed periodic decimal numbers to fractions

Pure period and mixed period decimal numbers are treated differently when converting to fractions. When converting mixed-period decimal numbers, you must first convert the mixed-period decimal number into a purely periodic decimal number by shifting the comma. To do this, you multiply the number by a suitable power of ten.

Important periodic decimal numbers as a fraction

Periodic decimal numbers for which you should know the representation as a fraction by heart: