Muggles' Guide to Harry Potter/Magic/Arithmancy

Overview
Arithmancy is a branch of magic involving the study of the magical properties of numbers. At Hogwarts, Arithmancy is an elective course, taught, we believe, by Professor Vector. It is offered to students in third to fifth years; those with sufficiently high O.W.L. marks in the course can then go on to N.E.W.T.-level study in years 6 and 7.

Extended Description
Hermione takes Arithmancy starting in her third year, and continues taking it, apparently, until her sixth year. We are told that she signs up for everything at the beginning of her third year, drops Divination at Easter, and then drops Muggle Studies at the end of the year. We also hear that she passed all her OWLs, which leads us to believe she has advanced to NEWT-level study in everything she chose to; and the only course we know she dropped at that point is Care of Magical Creatures. On the first day of her third year, Hermione says she is learning much more in Arithmancy than in Divination.

In Arithmancy class, a student is expected to understand complicated number charts and write essays, which is also part of a student's homework. This would seem to make it ideal for Hermione.

Analysis
In our world, Arithmancy is clearly related to numerology and gematria (Wikipedia link), belief in the mystical nature of numbers. We do not presume to judge whether there is any deeper meaning to numbers, but we suspect that the Wizarding world, working as it does with magic as an every-day thing, will have a better success rate with linking numbers to magical effects than we Muggles would.

We suspect that Tom Riddle's query, "isn't seven the most magically powerful number," in Harry Potter and the Half-Blood Prince, is based on something he learned in Arithmancy. This would, of course, be very basic arithmancy, probably first touched on in the first course year.

Quite separately from the series, a number of other fictional books about magic have cast doubt on the magical power of the number seven. One of the few that seems to have detailed the inner workings of magic as well as the Harry Potter series, the Lord Darcy series by Randall Garret, suggests that in fact seven is the least powerful number magically, because magic is based on the natural world, and the number seven is very rare in nature.