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Repeating decimals are not terminating. A repeating decimal continues on forever. One or more of its digits repeats continuously, and the decimal is infinite in length. Here are some examples:  Notice that even though the two's go on forever, you don't have to write all of them. (You couldn't!) There are several simple ways you can write an infinitely long decimal... You can either: - include three of the repeating digits, and then use three dots to show that it goes on forever - include just one repeating digit, and put a dot over that digit We'll be using both methods, so you'll become familiar with them. What happens if more than one digit repeats? Here's an example:  We wrote three repetitions of the '27' before using the three dots. Then, instead of a dot over the repeating digits, we used a bar over the repeating part. Of course, the repeating part can be any length at all. Here's an example with three repeating digits:  To be perfectly correct, we should have included three repetitions of the '145' before using the three dots. But this can get a little tedious if the repeating part is long, so sometimes you just repeat it twice. You must repeat it at least twice, so it is perfectly clear what is repeating. The bar method is a lot less work! It's also possible to have a decimal where part does not repeat, and part does. For example:  Look at this decimal very carefully. Notice that the '2' does not repeat. The '54' does ... and it's the part with the bar over it. Watch out for repeating decimals that are written incorrectly. For example, what decimal is this? 0.135... Does the entire '135' repeat? Or just the '5'? Nobody knows! That decimal was written incorrectly, and is a mistake. It should have been written as: 0.135135..., or perhaps 0.13555..., depending on what the person meant who wrote it. Better yet, write it with a dot over the repaeting digit(s). |