Arithmetic follows the "adventure" of three sisters after their kidnapping as they try to find their way home. However, their ages being 7, 5, and 3, they have little luck in terms of not only navigational skills but even general communication and logic. The story takes place in the late-1800's and is told from the perspective of the middle child (5 years old). While her older sister has some idea as to the fact that something bad is going on, she is pretty much oblivious. Each character is presented and known the way a girl of 5 years would know someone. As a result, characters are known not by names but by descriptive characteristics (the exception being Lady Kate and her two sisters).
Annie is the oldest sister and is very protective of her younger sisters. She is also the most developed mentally (for obvious reasons) and all sensible communication comes from her. Determined to return herself and her sisters to their home, she comes across as somewhat of a snob, ignoring people who feels can be of no help to her in her quest.
Claire is the narrator of the story. She is very imaginative and creative, but she is also quite clueless. She believes the whole situation to be some sort of terrific game. Right from the beginning she believed herself to be in a story of sorts, being shown a magical boat by fantastic magicians. As a result, she does not help at all in her older sisters attempts to get them back to their home.
Emily is the youngest sister. At only three years of age, she is more a burden than anything else. She tries very hard to keep up with her older sisters, but Claire having grown faster to match Annie, they are as good as a full four years ahead of her.