# 0 0 is a fraction

### Convert periodic decimal fractions to fractions

You know how to get from a fraction to a decimal fraction (divide numerator by denominator). If the division doesn't add up, you'll get periodic decimal fractions.

How does it work the other way around? How do you get from a periodic decimal fraction to the corresponding fraction?

Looking back: You can already convert non-periodic decimal fractions.

$$0,2=2/10=1/5$$

$$0,04=4/100=1/25$$

You convert instant-periodic decimal fractions by writing “9's numbers” in the denominator.

Convert $$ 0, \ bar (23) $$ to a fraction.

The period is 2 digits long. Your denominator is then 99. Your numerator is 23.

$$ 0, \ bar (23) = 23/99 $$

Another example:

$$ 0, \ bar (023) = 23/999 $$

This is how you convert instant-periodic decimal fractions into fractions: Write the period in the numerator and in the denominator as many nines as the period is long. Briefly if necessary. **example**:

$$ 0, bar (123) = 123/999 = 41/333 $$

### If you want to know more precisely why this works:

When you convert fractions whose denominator is nines, you find that you are getting the numerator as a period.

**Example 1:**

$$ 1/9 = 0, bar (1) $$

**Example 2:**

$$ 7/99 = 0. bar (07) $$

### Example $$ 0, \ bar (123) $$ examined more closely

Convert $$ 0, \ bar (123) $$ to a fraction.

Because the period **3** Digits is long, you take 1000 times the number:

$$ 0, \ bar (123) * 1000 = 123, \ bar (123) $$

You can easily subtract $$ 0, \ bar (123) $$ from this number. In both numbers, the same digits are repeated indefinitely after the decimal point.

If you subtract the number from the thousandfold of a number, you have $$ 999 $$ - times the number.

So you found out:

$$ \ 0, bar (123) * 999 = 123 $$

If you do the inverse problem, you get $$ \ 0, bar (123) = 123: 999 = 123/999 = 41/333 $$

In this way you have succeeded in converting the instant-periodic decimal number into a fraction.

You can use the same trick to convert any instantaneous decimal number to a **three-digit period** you get the digits of the period in the numerator and always $$ 999 $$ in the denominator.

### Convert mixed periodic decimal numbers

Converting mixed-periodic decimal fractions is unfortunately not that easy ...

That's how it's done:

Convert $$ 0.1bar (27) $$ to a fraction.

So that the period comes once before the decimal point and then repeats itself indefinitely after the decimal point, multiply by 1000:

$$ 0.1 \ bar (27) * 1000 = 127, bar (27) $$

You can only subtract an immediately periodic number from this number, i.e. not the number itself, but its tenfold:

$$ 0.1 \ bar (27) * 10 = 1, bar (27) $$.

With both numbers, the digits $$ 2 $$ and $$ 7 $$ after the decimal point are repeated infinitely often:

You can convert mixed-periodic decimal fractions by subtracting appropriate multiples from each other and then doing the inverse problem.

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### Another example

Convert $$ 0.01bar (6) $$ to a fraction.

So that the period comes once before the decimal point and then repeats itself indefinitely after the decimal point, multiply by 1000:

$$ 0.01bar (6) * 1000 = 16.bar (6) $$

You can only subtract an immediately periodic number from this number, i.e. not the number itself, but its hundredfold:

$$ 0.01bar (6) * 100 = 1.bar (6) $$. The $$ 6 $$ after the decimal point is repeated infinitely often for both numbers:

$$ 16.bar (6) = 0.01bar (6) * 1000 $$

$$ - $$ $$ 1, bar (6) = 0.01bar (6) * $$ $$ 100 $$

─────────────────

$$ 15 $$ $$ = 0.01bar (6) * $$ $$ 900 $$

So you get

$$ 0.01bar (6) = \ frac {15} {900} = \ frac {1} {60}. $$

##### Tip to check

In the denominator you get as many nines as the period is long and then as many zeros as there are digits between the comma and period.

### It continues

**example 1**: Convert $$ 0,0bar (1) $$ to a fraction.

Multiply by $$ 10 $$ and you get

$$ 10 * 0.0bar (1) = 0, bar (1) = 1/9 $$ and with the help of the inversion problem

$$ 0.0bar (1) = (1/9) / 10 = 1/90 $$.

**Example 2**: Convert $$ 0.00bar (1) $$ to a fraction.

Multiply by $$ 100 $$ and you get

$$ 100 * 0.0bar (1) = 0, bar (1) = 1/9 $$ and with the help of the inversion problem

$$ 0.00bar (1) = (1/9) / 100 = 1/900 $$.

**Example 3**: Convert $$ 0,0bar (01) $$ to a fraction.

Multiply by $$ 10 $$ and you get

$$ 10 * 0.0bar (01) = 0.bar (01) = 1/99 $$ and with the help of the inversion problem

$$ 0.0bar (01) = (1/99) / 10 = 1/990 $$.

### Put together

You can always write a mixed-periodic decimal number as the sum of a finite decimal number and a periodic decimal number

**Example 1:**

Convert $$ 2,4bar (3) $$ to a fraction.

Disassemble:

$$ 2.4bar (3) = 2.4 + 0.0bar (3) $$

The whole conversion:

$$ 2.4bar (3) = 2.4 + 0.0bar (3) = 2 4/10 + 3/90 = 2 12/30 + 1/30 = 2 13/30 $$

**Example 2:**

Convert $$ 0.08bar (3) $$ to a fraction.

$$ 0.08bar (3) = 0.08 + 0.00bar (3) = 8/100 + 3/900 = (24 + 1) / 300 = 25/300 = 1/12 $$

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