What areas of the brain are involved in doing arithmetic operations?
This is a part answer to your question.

According to "Arithmetic processing in the brain shaped by cultures" (Tang, 2006), the parts of the brain that are activated while doing arithmetic is to an extent, governed by culture, in the article, the authors assert:


  Using
  functional MRI, we demonstrated a differential cortical representation
  of numbers between native Chinese and English speakers.
  Contrasting to native English speakers, who largely employ a
  language process that relies on the left perisylvian cortices for
  mental calculation such as a simple addition task, native Chinese
  speakers, instead, engage a visuo-premotor association network
  for the same task. Whereas in both groups the inferior parietal
  cortex was activated by a task for numerical quantity comparison.


(The article also includes fMRI imagery examples).

It also seems that recognition of simple arithmetic starts at an early age, according to "Infant brains detect arithmetic errors" (Berger et al. 2006).