# Factorial!

Now and then you might see a number with an exclamation point after it, like this: $87!$. No, this is not a very excited number; it’s the factorial function. $n!$ (“n factorial”) means to multiply together all the integers from n down to 1. So, for example, $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$. The factorial function gets big pretty fast; for example, $10! = 10 \cdot 9 \cdot 8 \cdots 1 = 3,628,800$ (about 3.6 million), and $20! = 2,432,902,008,176,640,000$ (that’s 2.4 quintillion).

Note that $0!$ 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 $0!$ 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: $\begin{array}{rcl} 0! &=& 1 \\ n! &=& n \cdot (n-1)! \qquad \text{when } n \geq 1 \end{array}$

### 8 Responses to Factorial!

1. Pingback: Factorial! « Getzville LRC’s Weblog

2. emma says:

thank you so much for posting this. it helped with a project for school

3. Dave S says:

Did a dollar-latex go missing in the last edit of the recursive formula?

• Brent says:

Whoops, thanks for the heads up! I switched hosting software a while back and the LaTeX support changed slightly. I thought I had gone through and update everything but looks like I missed this.

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