When I was in school, perhaps my favorite pet project was trying to find a formula for the number of digits in . For starters:

: 1 digit

: 1 digit

: 1 digit

: 1 digit

: 2 digits

: 3 digits

: 3 digits

: 4 digits

: 5 digits

I owned what was then a top-of-the-line scientific calculator (with approximately the same computational capability as a modern TI-30), and I distinctly remember making a graph like the following on graph paper. The above calculations contribute the points , , , , , , , , and .

I had to stop (or, more accurately, I thought I had to stop) at because my calculator couldn’t handle numbers larger than .

I stared at this graph for weeks, if not months, trying to figure out an equation that would fit these points. And I never could figure it out.

And, to this day, I’m somewhat annoyed at my adolescent self that I wasn’t able to figure out this puzzle for myself… since I had all the tools in my possession needed to solve the puzzle, though I didn’t know how to use the tools.

In this series of posts, I’ll answer this question with the clever application of some concepts from calculus and precalculus.

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.
View all posts by John Quintanilla

Published

One thought on “The number of digits in n! (Part 1)”

## One thought on “The number of digits in n! (Part 1)”