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.
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