IEVref: 102-02-09 ID: Language: en Status: Standard Term: complex number Synonym1: Synonym2: Synonym3: Symbol: Definition: element of a set containing the real numbers and other elements, which may be represented by an ordered pair of real numbers (a, b), with following properties: the pair (a, 0) represents the real number a, an addition is defined by $\left({a}_{1}\text{,}\text{\hspace{0.17em}}{b}_{1}\right)+\left({a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{2}\right)=\left({a}_{1}+{a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{1}+{b}_{2}\right)$, a multiplication is defined by $\left({a}_{1}\text{,}\text{\hspace{0.17em}}{b}_{1}\right)×\left({a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{2}\right)=\left({a}_{1}{a}_{2}-{b}_{1}{b}_{2}\text{,}\text{\hspace{0.17em}}{a}_{1}{b}_{2}+{a}_{2}{b}_{1}\right)$ Note 1 to entry: All properties of real numbers (operations and limits) are extended to complex numbers except the order relation.Note 2 to entry: The complex number defined by the pair (a, b) is denoted by $c=a+jb$ where j is the imaginary unit (IEV 102-02-10) represented by the pair (0, 1), a is the real part and b the imaginary part. A complex number may also be expressed as $c=|c|\left(\mathrm{cos}\phi +j\text{\hspace{0.17em}}\mathrm{sin}\phi \right)=|c|{e}^{j\phi }$ where $|c|$ is a non-negative real number called modulus and φ a real number called argument. Note 3 to entry: In electrotechnology, a complex number is usually denoted by an underlined letter symbol, for example $\underset{_}{c}$. Note 4 to entry: The set of complex numbers is denoted by ℂ (C with a vertical bar in the left arc) or C. This set without zero is denoted by an asterisk to the symbol, for example ℂ*. Publication date: 2008-08 Source: Replaces: Internal notes: 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO CO remarks: TC/SC remarks: VT remarks: Domain1: Domain2: Domain3: Domain4: Domain5: