Glossary—Numbers

Imaginary Basis The basis of the imaginary numbers is \({\mathrm{j}},\) with \({\mathrm{j}}^{2}=-1.\) This notation follows the engineering conventions.

Signum Function Let \({\mathrm{j}}\) be the basis of the imaginary numbers. Then, the signum of a number is defined as the complex function \({\mathrm{sgn}}:{\mathbb{C}}\rightarrow{\mathbb{C}}\). It is such that \(z\mapsto{\mathrm{sgn}}\,z=\left\{\begin{array}{ll}0,&z=0\\{\mathrm{e}}^{{\mathrm{j}}\,\arg z},&z\neq0.\end{array}\right.\)

Negative Numbers The complex number \(z\in{\mathbb{C}}\) is said to be negative when \({\mathrm{sgn}}\,z=-1.\) It is said to be nonnegative otherwise. When the domain of the signum function is restricted to the real numbers, negative numbers are written \(\{x\in{\mathbb{R}}|x<0\}\) in set notation, \((-\infty,0)\) in interval notation, and \({\mathbb{R}}_{<0}\) for short. When the domain is restricted to the integers, the negative integers are written \({\mathbb{Z}}\setminus{\mathbb{N}}={\mathbb{Z}}_{<0}=\left(-{\mathbb{N}}\right)\setminus\{0\}\) while the nonnegative integers are written \({\mathbb{N}}={\mathbb{Z}}_{\geq0}.\) The number zero is neither negative nor positive.

Positive Numbers The complex number \(z\in{\mathbb{C}}\) is said to be positive when \({\mathrm{sgn}}\,z=1.\) It is said to be nonpositive otherwise. When the domain of the signum function is restricted to the real numbers, positive numbers are written \(\{x\in{\mathbb{R}}|x>0\}\) in set notation, \((0,\infty)\) in interval notation, and \({\mathbb{R}}_{>0}\) for short. When the domain is restricted to the integers, the positive integers are written \({\mathbb{N}}\setminus\{0\}=\left({\mathbb{N}}+1\right)={\mathbb{Z}}_{>0}\) while the nonpositive integers are written \(\left(-{\mathbb{N}}\right)={\mathbb{Z}}_{\leq0}=\left({\mathbb{Z}}\setminus{\mathbb{N}}\right)\cup\{0\}={\mathbb{Z}}\setminus\left({\mathbb{N}}\setminus\{0\}\right).\) The number zero is neither negative nor positive.

Even Numbers The even numbers are notated \(\left(2\,{\mathbb{Z}}\right).\) The nonnegative even numbers are notated \(\left(2\,{\mathbb{N}}\right).\) The positive even numbers are notated \(\left(2\,{\mathbb{N}}+2\right).\)

Odd Numbers The odd numbers are notated \(\left(2\,{\mathbb{Z}}+1\right).\) The positive odd numbers are notated \(\left(2\,{\mathbb{N}}+1\right).\)