I think arithmancy is about investigating the magical science of numbers and whatnot. I can see why it would appeal to Hermione.
Arithmancy is an ancient form of divination, but instead of predicting the future by interpreting fuzzy images and squiggles you use hard-and-fast rules and mathematical calculations.
Arithmancy, from the Greek arithmo meaning “number” and mancy meaning “prophecy”, has been used by witches and wizards for more that 2000 years to help people analyze and develop their strengths and talents, over come obstacles and chart their future paths.
It is also known as “numerology” and is based on two ideas. The first, that a persons name contains important clues to the character and destiny. The second, advanced more than 2500 years ago by the Greek sage Pythagoras, is that each of the numbers between 1 and 9 has a unique meaning that can contribute to the understanding of all things.
These two ideas were combined over the centuries developing many complex systems form converting names into numbers and analyzing the result. The most commonly used systems involves extracting three key numbers from a persons name (The Character Number, the Heart Number and the Social Number) and interpreting the outcome according to a set of established meanings. This system, taught at Hogwarts, was widely known by the Middle Ages and is still used today. All you need is a pencil and paper and to know how to add and spell.
Convert the name to a set of numbers, each letter of the alphabet has a number between 1 and 9, as follows:
1 2 3 4 5 6 7 8 9
A B C D E F G H I
J K L M N O P Q R
S T U V W X Y Z
To analyze any name, write it down with the corresponding numerical value. We're using Nicholas Flamel as an example:
N I C H O L A S F L A M E L
5 9 3 8 6 3 1 1 6 3 1 4 5 3
When you have all the numbers written down, add the up. With Nicholas Flamel, the result is 58. According to the procedures of Arithmancy, when the total is more than 9 (which it almost always is) it must be made a single digit by adding the numbers together until this is achieved. 58 becomes 13 (5 + 8 = 13), which becomes 4 (1 + 3 = 4).
This number, whatever it is for your name, is known as the Character Number. It indicates the general personality type of the individual.
The next number is the Heart Number which refers the persons inner life and is said to indicate desires and fears hidden from others, this is the total off all the vowels in the name.
N I C H O L A S F L A M E L
9 6 1 1 5
This equals 22 whuch equals 4 (2 + 2 = 4). In Nicholas Flamels' case the heart and character number are the same, but it is not always like that.
The third number is the Social Number, which refers to the outer personality, the face a person shows to the outside world. It is the value of the consonants in the name.
N I C H O L A S F L A M E L
5 3 8 3 1 6 3 4 3
This equals 36 which equals 9 (3 + 6 = 9).
With the numbers in mind (4, 4, 9) it is now possible to sketch out a portrait of the subject using a set of positive and negative personality traits traditionally associated with each number.
The Meaning of the Numbers
ONE: This is the number of the individual. One's are independent, focused, single-minded and determined. They set a goal and stick to it. They are the leaders and inventors. One's find it difficult to work with others and don't like to take orders. They can be self-centred, egotistical and domineering. They are often loners.
TWO:Two represents interaction, two-way communication, co-operation and balance. Two's are imaginative, creative and sweet natured. Peace, harmony, commitment, loyalty and fairness are characteristics. Two also introduces the idea of conflict, opposing forces and the contrasting sides of things: night and day, good and evil. Two's can be withdrawn, moody, self-conscious and indecisive.
THREE: Three represents the idea of completeness or wholeness, as in the trio's “past-present-future” and “mind-body-spirit.” The Pythagoreans considered three to be the first “complete” number because, like three pebbles laid out in a row, it has a beginning, a middle and an end. Three indicates talent, energy, an artistic nature, humour and social ease. Three's are often lucky, easy going, rich and highly successful, but they can also be unfocused, easily offended and superficial.
FOUR: Like a table that rests solidly on four legs, four indicates stability and firmness. Four's enjoy hard work. They are practical, reliable and down to earth; they prefer logic and reason to flights of fancy. They are good at organising and getting things done. Like the cycle of the four seasons, they are also predictable. They can be stubborn, suspicious, over practical and prone to angry outbursts. The conflicts possible with “two” are doubled in four.
FIVE: Five is the number of instability and imbalance, indicating change and uncertainty. Five are drawn to many things at once but commit to none. They are adventurous, energetic and willing to take risks. They enjoy travel and meeting new people but may not stay in one place very long. Five's can be conceited, irresponsible, quick-tempered and impatient.
SIX: Six represents harmony, friendship and family life. Sixes are loyal, reliable and loving. They adapt easily. They do well in teaching and the arts, but are often unsuccessful in business. They are sometimes prone to gossip and complacency. The Pythagoreans regarded six and the perfect number because it was divisible by both two and tree, and was the sum as well as the product of the first three digit. (1 + 2 + 3 = 6, 1 x 2 x 3 = 6)
SEVEN: Perceptive, understanding and bright, sevens enjoy hard work and challenges. They are often serious, scholarly, and interested in all things mysterious. Originality and imagination are more important than money and material possessions. Sevens can also be pessimistic, sarcastic and insecure. Seven is sometimes considered a mystical or magical number because of its associations with the biblical seven days of creation and the seven heavenly bodies of ancient astronomy (Sun, Moon, Mercury, Venus, Mars, Saturn and Jupiter).
EIGHT: Eight indicates the possibility of great success in business, finance and politics. Eights are practical, ambitious, committed and hardworking. They can also be jealous, greedy, domineering and power hungry. Eight is said to be the most unpredictable of numbers and can indicate the pinnacle of success or the depths of failure; the potential to go either way is present from the beginning.
NINE: Represents completion and achievement to the fullest degree, as it is the “complete” number, three, expressed three times (3 x 3 = 9). Nines dedicate themselves to the service of others, often as teachers, scientists and humanitarians. Strongly determined, they work tirelessly and are and inspiration to others. However, they can also be arrogant and conceited when things don't go their way.
By this we can see that according to Nicolas Flamels' numbers, (4, 4, 9), he is a hardworking, down-to-earth person. He takes a practical approach to solving problems. He is emotionally stable, but may have some inner anger and suspicions; however, the ace he shows to the world is that of a kind and generous humanitarian. Finally, we can add a few finishing touches to this portrait by returning to the original array of numbers, to see if any digits occur more frequently than others. In this case the numbers 3 and 1 occur most often indicating that, in addition to what we already know, Flamel is someone who seeks perfection, has the ability to make money easily and goes about things with single-minded focus. Most of this is true about Nicholas Flamel.
Arithmancy can also be applied to other things in life, almost anything really. An 8 personality for example would be advised to make important decisions on the 8th, 17th or 26th and a 5 on the 5th, 14th or 23rd and so on. It can be used to work out hidden friendships between people or things. A 6, for example, will be happiest driving a Honda or Toyota and a 7 happiest driving a Ford. Five's would be happiest living in Tokyo or Pittsburgh and so on.
As you can see, Arithmancy can be applied to anything in life and is much more accurate than Divinations and Tea-Leaf reading which is probably why Hermione likes it so much.