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IEVref: | 102-02-09 | ID: | |

Language: | en | Status: Standard | |

Term: | complex number | ||

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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 $({a}_{1}\text{,}\text{\hspace{0.17em}}{b}_{1})+({a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{2})=({a}_{1}+{a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{1}+{b}_{2})$,
- a multiplication is defined by $({a}_{1}\text{,}\text{\hspace{0.17em}}{b}_{1})\times ({a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{2})=({a}_{1}{a}_{2}-{b}_{1}{b}_{2}\text{,}\text{\hspace{0.17em}}{a}_{1}{b}_{2}+{a}_{2}{b}_{1})$
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 ( 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 | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO | ||

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- the pair (
*a*, 0) represents the real number*a*, - an addition is defined by $({a}_{1}\text{,}\text{\hspace{0.17em}}{b}_{1})+({a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{2})=({a}_{1}+{a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{1}+{b}_{2})$,
- a multiplication is defined by $({a}_{1}\text{,}\text{\hspace{0.17em}}{b}_{1})\times ({a}_{2}\text{,}\text{\hspace{0.17em}}{b}_{2})=({a}_{1}{a}_{2}-{b}_{1}{b}_{2}\text{,}\text{\hspace{0.17em}}{a}_{1}{b}_{2}+{a}_{2}{b}_{1})$

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 $c=\left|c\right|(\mathrm{cos}\phi +j\text{\hspace{0.17em}}\mathrm{sin}\phi )=\left|c\right|{e}^{j\phi}$ where $\left|c\right|$ 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*.