AI Tutor
If a decimal goes on forever without stopping, like 0.333…0.333\dots, how can it possibly equal a clean fraction like 13\frac{1}{3}?
It turns out there is an algebraic trick that cancels out the infinite tail completely.

The Infinite Tail Trick

A rational number is any number you can write as a ratio ab\frac{a}{b}, where aa and bb are integers and b≠0b \neq 0.
To convert a repeating decimal into a fraction, we name the decimal xx, multiply by a power of 10 to shift the decimal point, and subtract the two equations to erase the repeating part.
Let's walk through 0.777…0.777\dots step by step.
First, set x=0.777…x = 0.777\dots Since only one digit repeats, multiply both sides by 1010 to get 10x=7.777…10x = 7.777\dots
Now subtract the original equation from the new one: 10xβˆ’x=7.777β‹―βˆ’0.777…10x - x = 7.777\dots - 0.777\dots, which gives 9x=79x = 7, or x=79x = \frac{7}{9}.
What happens when more than one digit repeats in the pattern?

Two Repeating Digits

When a block of two digits repeats, like in 0.818181…0.818181\dots, multiplying by 1010 gives 8.1818…8.1818\dots, which doesn't align with the original digits.
Instead, multiply by 100100 (one zero for every repeating digit) so the repeating pattern lines up perfectly behind the decimal point.