AI Tutor
How can an infinite string of numbers like 0.7777…0.7777\dots shrink into a simple fraction like 79\frac{7}{9}?
It turns out you can destroy an infinite decimal tail with a single subtraction step.

Single Repeating Digit

Let's assign a variable to our decimal: let x=0.777…x = 0.777\dots (also written as 0.7Λ‰0.\bar{7}).
Because exactly one digit repeats, multiply both sides by 101=1010^1 = 10 to slide the decimal point one place to the right.
This gives 10x=7.777…10x = 7.777\dots. When you subtract the original equation from this new one, the infinite tails match perfectly and cancel out to zero.
10xβˆ’x=7.777β‹―βˆ’0.777…10x - x = 7.777\dots - 0.777\dots 9x=7β€…β€ŠβŸΉβ€…β€Šx=799x = 7 \implies x = \frac{7}{9}
What if the repeating cycle has two, three, or more digits repeating in a loop?

Multi-Digit Repeating Blocks

The length of the repeating cycle is called the period. When a number has a period of nn, multiply by 10n10^n so the decimal tails line up.
Let's convert 0.54β€Ύ=0.545454…0.\overline{54} = 0.545454\dots. Since two digits repeat (n=2n=2), multiply xx by 102=10010^2 = 100.