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If you divide 11 by 33, the decimal 0.333…0.333\dots goes on forever without stopping. Yet 11 divided by 33 is an exact fraction.
So why are some infinite decimals clean fractions, while numbers like Ο€\pi or 2\sqrt{2} can never be written as a fraction at all?

Rational Numbers and Their Decimals

A rational number is any number that can be written as a fraction ab\frac{a}{b}, where aa and bb are integers and the denominator bb is not zero.
When you convert a rational number into a decimal by dividing, you only get two possible behaviors: the decimal either terminates or repeats.
A terminating decimal stops after a finite number of digits, like 0.75=340.75 = \frac{3}{4}, while a repeating decimal repeats a digit or block of digits infinitely, written with a bar called a vinculum like 0.3β€Ύ0.\overline{3}.
Converting a terminating decimal back to a fraction is simple place value β€” but what about repeating decimals?

Converting Repeating Decimals to Fractions

To turn a repeating decimal into a fraction, we use an algebraic trick: we multiply by a power of 1010 to shift the repeating pattern, then subtract the original decimal to cancel out the infinite tail.
Let us convert 0.7β€Ύ0.\overline{7} into a fraction by setting x=0.777…x = 0.777\dots and multiplying by 1010.
If two digits repeat, such as x=0.45β€Ύx = 0.\overline{45}, multiply by 100100 instead: 100x=45.45β€Ύ100x = 45.\overline{45}, so 100xβˆ’x=45100x - x = 45, which gives 99x=4599x = 45 or x=4599=511x = \frac{45}{99} = \frac{5}{11}.