Now and then you might see a number with an exclamation point after it, like this: . No, this is not a very excited number; it’s the factorial function. (“n factorial”) means to multiply together all the integers from n down to 1. So, for example, . The factorial function gets big pretty fast; for example, (about 3.6 million), and (that’s 2.4 quintillion).
Note that is defined to be 1. This might seem a little strange until you consider the fact that there aren’t any integers “from 0 down to 1”, so really means to multiply together no integers. If you go to multiply some integers but find that you don’t have any to multiply, you get 1, since 1 is the multiplicative identity. (For a more detailed discussion of this, try reading the Empty product article on Wikipedia.)
Notice that factorial has an elegant recursive definition: