ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Collection of sets
Set union
Successor set
Inductive set
Set of inductive sets
Set of natural numbers
Set of integers
Integer
Integer base-2 representation
N-bit integer
Definition D6266
16-bit integer
Formulation 0
An D995: Integer $a \in \mathbb{Z}$ is a 16-bit integer if and only if \begin{equation} \exists \, r_0, r_1, \ldots, r_{15} \in \{ 0, 1 \} : a = - r_{15} 2^{15} + \sum_{n = 0}^{14} r_n 2^n \end{equation}
Formulation 1
An D995: Integer $a \in \mathbb{Z}$ is a 16-bit integer if and only if \begin{equation} \begin{split} \exists \, r_0, r_1, \ldots, r_{15} \in \{ 0, 1 \} : a & = r_0 + 2 r_1 + 4 r_2 + 8 r_3 + 16 r_4 + 32 r_5 + 64 r_6 + 128 r_7 \\ & \quad + 256 r_8 + 512 r_9 + 1024 r_{10} + 2048 r_{11} + 4096 r_{12} \\ & \quad + 8192 r_{13} + 16384 r_{14} - 32768 r_{15} \end{split} \end{equation}