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. Pingback: Irrationality of pi: the unpossible function « The Math Less Traveled

  3. emma says:

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

  4. Pingback: Irrationality of pi: the unpossible function | The Math Less Traveled

  5. Pingback: Binomial coefficients | The Math Less Traveled

  6. 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.

  7. Pingback: Permuting permutations | The Math Less Traveled

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