This common question arises because does not fit the usual definition for . Recall that, for positive integers, we have

Going from the bottom line to the top, we see that start at , and then multiply by , then multiply by , then multiply by , then multiply by . To get , we multiply the top line by :

.

Because they’re formed by successive multiplications, the factorials get large very, very quickly. I still remember, years ago, writing lesson plans while listening to the game show Wheel of Fortune. After the contestant solved the final puzzle, Pat Sajak happily announced, “You’ve just won $40,320 in cash and prizes.” My instantaneous reaction: “Ah… that’s .” Then I planted a firm facepalm for having factorials as my first reaction. (Perhaps not surprisingly, I was still single when this happened.)

Back to . We can also work downward as well as upward through successive division. In other words,

divided by is equal to .

divided by is equal to .

divided by is equal to .

divided by is equal to .

Clearly, there’s one more possible step: dividing by . And so we define to be equal to divided by , or

.

Notice that there’s a natural way to take another step because division by 0 is not permissible. So we can define , but we can’t define .

In Part 2, I’ll present a second way of approaching this question.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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