A calculator shows 0.333333, and a student writes that one third is “almost” that number. The display is limited, but the fraction itself is exact. An endlessly repeating decimal and a decimal cut off after a few places are different things.
Write 0.333… to mean that the threes continue without ending. The dots matter: 0.33 is exactly thirty-three hundredths, while 0.333… represents one third.
Use subtraction to remove the repeating tail
Let x = 0.333…. Multiplying by ten gives 10x = 3.333…. Subtract the first equation from the second. The identical repeating tails cancel, leaving 9x = 3. Therefore x = 3/9 = 1/3.
The calculation works because the repeated part lines up. It is not a method for ignoring inconvenient digits; the two infinite tails are the same after the shift.
Try a two-digit pattern next. If y = 0.272727…, then 100y = 27.272727…. Subtracting gives 99y = 27, so y = 27/99 = 3/11. Multiplying by one hundred moves one complete two-digit repeat.
Check what the notation is claiming
A line drawn over the repeating digits can make the pattern clearer than dots. Place it over the whole repeated block, not over a digit that appears only once before repetition begins.
For practice, ask a learner to sort three expressions: 0.27, 0.2727 and 0.272727…. The first two terminate and can be written over powers of ten. The last repeats indefinitely and needs a different exact fraction.
Finish by dividing the proposed fraction and checking the pattern. A calculator can illustrate the result, but its last displayed digit may be rounded. The algebra explains the exact relationship; the screen gives only a finite view of it.