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

In Part 1 of this series, I discussed descending down this lines with repeated division to define .

Here’s a second way of explaining why that may or may not be as convincing as the first explanation. Let’s count the number of “words” that can made using each of the three letters A, B, and C exactly once. Ignoring that most of these don’t appear in the dictionary, there are six possible words:

ABC, ACB, BAC, BCA, CAB, CBA

With two letters, there are only two possible words: AB and BA.

With four letters, there are 24 possible words:

ABCD, ABDC, ACBD, ACDB, ADBC, ADCB,

BACD, BADC, BCAD, BCDA, BDAC, BDCA,

CABD, CADB, CBAD, CBDA, CDAB, CDBA,

DABC, DACB, DBAC, DBCA, DCAB, DCBA.

Evidently, there are different words using four letters, different words using three letters, and different words using two letters.

Why does this happen? Let’s examine the case of four letters. First, there are different possible choices for the first letter in the word. Next, the second letter can be anything but the first letter, so there are different possibilities for the second letter. Then there are remaining possibilities for the third letter, leaving possibility for the last.

In summary, there are , or , different possible words. The same logic applies for words formed from three letters or any other number of letters.

What if there are 0 letters? Then there is only 1 possibility: not making any words. So it’s reasonable to define .

It turns out that there’s a natural way to define for all complex numbers that are not negative integers. For example, there’s a reasonable way to define , and even . I’ll probably discuss this in a future post.