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Periodic decimal fractions

You have already calculated a lot with fractions and decimal fractions, but there is another special feature: periodic decimal fractions. You have an infinite number of decimal places!

But one after anonther…

There are 2 ways to convert a fraction to a decimal fraction.

Way 1: expand or shorten

You can easily convert a fraction with a power of ten in the denominator:

example 1:

$$ 4/5 \ stackrel (2) = 8/10 = 0.8 $$

Example 2:

$$ 7/40 \ stackrel (25) = 175/1000 = 0.175 $$

Way 2: Fraction as a quotient

You can write each fraction as a division problem and then do the math.

example:

$$7/40=7:40$$

This is how you walk one Fraction into a decimal fraction around:
Expand or shorten until you have a power of ten in the denominator. The decimal fraction has as many decimal places as the denominator has zeros.

You can write any fraction as a quotient of $$ 2 $$ natural numbers.

$$ \ text (numerator) / \ text (denominator) = \ text (numerator): \ text (denominator) $$

What if the division doesn't open?

Well, then take a look at this example:

It goes on and on! At the top there is always the 6 and at the bottom there is always the remaining 2.

These are periodic decimal fractions. They are called periodically because a sequence of digits (here the 6) is repeated over and over again. The period is the 6th

You write: $$ 2/3 = 0, \ bar (6) $$

You mark the period with a line.

You say: "Zero point period $$ 6 $$"

If you take a fraction as a quotient and a digit or group of digits is repeated over and over, you get a periodic decimal fraction. Example: $$ 2/3 = 0.66666… = 0, \ bar (6) $$

Say: "Zero point period $$ 6 $$"

The word "periodic" comes from "Periodos" (Greek): going around, circulating, returning

Multiple digits in the period

Another example:

Convert $$ 6/11 $$ to a decimal fraction.

Two digits are repeated here, not just one.

You write it like this: $$ 6/11 = 0, bar (54) $$

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Mixed-periodic decimal numbers

It can also happen that the period does not start immediately after the decimal point, but only later.

Convert $$ 5/6 $$ to a decimal fraction.

The period only comes after the 8.

Put the dash for the period exactly above the 3, not the 8.

$$ 5/6 = 0.8bar3 $$

Say: "Zero point 8 period 3"

Examined in more detail

How long can such a period be?

Example: $$ 3/7 $$


So: $$ 3/7 = 0, \ bar (428571) $$

With a 7 in the denominator, the maximum period length is 6 (7–1).

All six possible residues when dividing by $$ 7 $$ have occurred. The period of the floating point number belonging to $$ 3/7 $$ cannot be longer than six digits.

The period can also be shorter than the maximum length.

This is the case with $$ 2/3 = 0, \ bar (6) $$. The period consists of just one digit instead of $$ 2 = 3-1 $$.

Numbers games

Not all decimal fractions with an infinite number of decimal places are periodic.

Examples:

  • $$0,1101001000…$$
  • $$0,12345678911223344…$$
  • You can think of other examples.

Infinite decimal numbers that are not periodic cannot be written as fractions.

With a fraction there is always a period when the division does not work. This is because there are only as many different residues possible as the denominator indicates. At some point the result numbers have to repeat themselves.

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Summary

There are three types of point numbers associated with fractions:

  1. finite or terminating decimal fractions

    $$ 1/5 \ stackrel (2) = 2/10 = 0.2 $$

  2. immediately periodic or purely periodic decimal fractions

    $$ 8/9 = 0, \ bar (8) $$

  3. mixed-periodic decimal fractions

    $$ 5/6 = 0.8 \ bar (3) $$

Every fraction has a finite or periodic decimal fraction and vice versa.