So far, all the decimal arithmetic we've done has involved decimal numbers with a finite number of decimal places. However, sometimes decimal numbers are infinite. Make sure you don't confuse "infinite decimal" with "infinitesimal." Although a number can sometimes be both, they're not the same thing. Even though they sound exactly the same when pronounced aloud. Thanks again, English.
Infinite decimals sometimes show up when we convert fractions into decimals.
Convert the fraction 1/3 into a decimal, using long division:
We end up with a decimal that goes on forever. Literally. And we thought Mondays seemed long.
To show an infinite decimal, we write "..." at the end. This is also good for when you get bored writing all the digits of a lengthy finite decimal, or when your pen is running out of ink.
Another way to write an infinite decimal with a repeating pattern is to draw a bar over the part that repeats.
0.333333333.... = .3
(On a side note, if one bar isn't enough and you wanted to completely cage in that 3, you could make him the numerator in a fraction and take the absolute value: . There. He's not going anywhere now.)
There are also infinite decimals without repeating patterns. These decimals represent the irrational numbers, and there is no way to know all the digits of any such number. And please don't take that as a personal challenge. We don't want to see you wasting the next thirty years of your life trying to memorize pi to an infinite number of digits before finally realizing it can't be done. We'd feel partly responsible.
However, you can see at least the first few digits of some famous infinite decimals: