AI Tutor
If you punch 1Γ·41 \div 4 into a calculator, it stops cleanly at 0.250.25, but punch in 1Γ·31 \div 3 and you get an endless march of 0.3333…0.3333\dots
Why do some fractions stop while others loop forever, and is every repeating decimal actually a hidden fraction?

What Makes a Number Rational?

A rational number is any number that can be written as a ratio ab\frac{a}{b}, where aa and bb are integers (whole numbers or their negatives) and b≠0b \neq 0.
The word "rational" comes from the word ratio β€” if you can write it as one integer divided by another, it belongs to this family.
How do we know whether a fraction will stop or repeat forever when we divide it out?

Fractions to Decimals

To turn any fraction ab\frac{a}{b} into a decimal, divide the top number aa by the bottom number bb using long division.
A terminating decimal ends because the division hits a remainder of zero, like 38=0.375\frac{3}{8} = 0.375.
A repeating decimal never ends because the remainders start looping, creating a repeating block of digits indicated with a top bar (a vinculum), like 411=0.3636β‹―=0.36β€Ύ\frac{4}{11} = 0.3636\dots = 0.\overline{36}.
Converting fractions to decimals is straightforward, but what if you are handed a decimal and need to work backwards to find its fraction?