transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Arithmetic (Greek ἀριθμός: number) is, roughly speaking, the study of numbers in their various forms, and the structure and properties of the operations defined on them, including at least addition and multiplication, and sometimes also subtraction, division, and exponentiation.
Notions of “number” are very broad and not at all easy to encapsulate. There are natural numbers, integers, rational numbers, real numbers, and complex numbers, quaternions (Hamiltonian numbers), octonions (Cayley numbers). There are dual numbers and hyperbolic numbers?, and geometric algebras (geometric numbers). There are integers modulo $n$. There are algebraic numbers and algebraic integers, and individual fields of such (number fields). There are $p$-adic numbers. Then there are cardinal numbers, ordinal numbers, and surreal numbers. For each one of these one can (and does!) speak of its arithmetic.
This article will provide links to other articles in which these various cases are discussed.
See also