Welcome to Mastery of Repeating Decimals! This is the first lesson of the course, and over the next four lessons we will build the skills needed to read, write, analyze, and convert repeating decimals with confidence. As you may recall from earlier work with rational numbers, every rational number's decimal expansion either terminates or eventually falls into a repeating cycle. We have already spent time on the terminating side, so now it is time to turn our full attention to the repeating kind.

In this lesson, we will learn to read and write bar notation, the compact and precise way mathematicians represent repeating decimals. By the end, you will be able to move fluently in both directions: from bar notation to a written-out string of digits, and from a string of digits back to bar notation.

When you divide $1$ by $3$, the result is $0.3333\ldots$, where the digit $3$ continues forever. Writing a handful of digits followed by dots works in casual settings, but it can be ambiguous. For instance, does $0.16666\ldots$ mean only the $6$ repeats, or is the pair repeating? Without a precise convention, the reader is left guessing.

The horizontal line used in bar notation is called an overline (sometimes called a vinculum). It sits directly above the digit or group of digits that cycles endlessly. We call those digits the repeating block (also known as the repetend).

Here are three quick examples showing how the overline works:

Not every repeating decimal starts its cycle immediately after the decimal point. Sometimes one or more digits appear first and never repeat. We call those leading digits the non-repeating prefix. The overline covers only the part that truly repeats.

Consider $\frac{1}{6}$, whose decimal expansion is $0.1666\ldots$ Here the digit $1$ shows up just once, while repeats forever. In bar notation we write:

So far we have been reading bar notation and expanding it into digits. Now let's practice going the other way: given a decimal written out with enough digits to reveal the pattern, we need to identify the repeating block and place the bar precisely.

A simple three-step approach works well:

1. Write out enough digits to spot which group of digits repeats.
2. Identify any digits that come before the repeating block (the non-repeating prefix, if any).
3. Place the overline above exactly the repeating block — nothing more, nothing less.

For example, if you see $0.454545\ldots$, the pair 45 clearly cycles and there is no prefix, so we write $0.\overline{45}$. If instead you see $0.8333\ldots$, the digit appears once and then repeats, giving us .

The exact placement of the overline changes the meaning of the number completely. Mixing up which digits fall under the bar is the most common mistake when working with this notation. Compare these two expressions side by side:

In this lesson, you learned that bar notation uses an overline to mark the exact digits that repeat in a decimal expansion. We saw how to unfold that notation into an infinite string of digits, how to spot a repeating block in a written-out decimal and write the matching bar notation, and why precise placement of the overline matters — shifting it by even one digit creates an entirely different number.

Now it is time to put these skills into action. In the practice exercises ahead, you will match bar notation expressions to their digit expansions, unfold repeating blocks into specific digit sequences, and write your own bar notation from written-out decimals. Let's jump in!