transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Given a star-algebra $(A ,\ast)$ in characteristic zero, then for $a\in A$ any element
its real part is $Re(a) \coloneqq \frac{1}{2}(a + a^\ast)$;
its imaginary part is $Im(a) \coloneqq \frac{1}{2}(a - a^\ast)$,
where $(-)^\ast$ is the given star-anti-involution (e.g. complex conjugation, quaternionic conjugation, …).
Standard situations where real and imaginary parts are considered include the real normed division algebra
$A$ the complex numbers;
$A$ the quaternions;
$A$ the octonions
Last revised on January 26, 2021 at 12:19:33. See the history of this page for a list of all contributions to it.