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 manifold 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, complex numbers, quaternions (Hamiltonian numbers), octonions (Cayley numbers). 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
Last revised on May 24, 2017 at 10:21:05. See the history of this page for a list of all contributions to it.