Question

how do you show a group of digits that repeat in a repeating decimal

Answer 1

In a repeating decimal, a group of digits that repeat can be indicated using a bar notation (also known as a vinculum) or parentheses. Below are the methods to represent repeating decimals:

1. **Bar Notation**: You place a horizontal bar over the digits that repeat. For example:
- The decimal \(0.333...\) (where the digit 3 repeats) can be written as \(0.\overline{3}\).
- The decimal \(0.142857142857...\) (where the sequence 142857 repeats) can be written as \(0.\overline{142857}\).

2. **Parentheses**: You can also use parentheses to indicate the repeating part:
- The decimal \(0.333...\) can be represented as \(0.(3)\).
- The decimal \(0.142857142857...\) can be expressed as \(0.(142857)\).

Both methods clearly indicate which part of the decimal is repeating. In formal mathematics, the bar notation is more commonly used, but both methods are understood and acceptable.