Welcome back to Mastery of Repeating Decimals! In the first lesson, we learned how bar notation works: the overline marks the exact digits that cycle, any digits before the bar form a non-repeating prefix, and even a small shift in bar placement changes the number entirely. With that foundation in place, you are ready for this second lesson.

Here we sharpen one specific skill: looking at a decimal expansion written out with trailing dots and figuring out exactly which digits repeat and which do not. This is trickier than it sounds, especially when several non-repeating digits appear before the cycle kicks in. By the end, you will be able to decompose any mixed repeating decimal into its two parts and write the correct bar notation with confidence.

In this lesson, we focused on decomposing mixed repeating decimals into their non-repeating prefix and repeating block. We built a reliable four-step strategy — write enough digits, read from the tail, find the shortest cycling string, and mark everything before it as the prefix — and we saw why expanding your bar notation back into digits is the simplest way to catch errors.

Up next, you will put this skill to work in a set of hands-on practice exercises. You will identify non-repeating and repeating parts in mixed decimals, fill in decomposition statements, and write precise bar notation for increasingly tricky expansions. Let's see how sharp your pattern-spotting eye has become!