• the pair (a, 0) represents the real number a,
• an addition is defined by $(a_1\text{,}{b}_1)+(a_2\text{,}{b}_2)=(a_1+a_2\text{,}{b}_1+b_2)$,
• a multiplication is defined by $(a_1\text{,}{b}_1)\times (a_2\text{,}{b}_2)=(a_1{a}_2-b_1{b}_2\text{,}{a}_1{b}_2+a_2{b}_1)$

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 $c=\left|c\right|(\cos \phi +j\sin \phi )=\left|c\right|e^{j\phi }$ 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*.