# What is 0 1 9

### 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 \$\$

How to 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) \$\$.

The digits \$\$ 2 \$\$ and \$\$ 7 \$\$ after the decimal point are repeated infinitely often for both numbers: 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) \$\$. For both numbers, the \$\$ 6 \$\$ after the decimal point is repeated infinitely often:

\$\$ 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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