Cell E3 contains a date like this: 6/5/06
Cell E4 is supposed to show how many days have passed since that
date... so here's my formula:
=TODAY()-E3

The result is 1/20/1900

What am I doing wrong exactly?

I'm going to take a bunch of entries and average out the number of days
that have passed from each... but I can't seem to get an integer... i
just get an obscure date.