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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