# Complex Variables

Properties of Complex Numbers. The symbol for a complex number *z* is *z* = *x* + *iy*, where *x* and *y* are real numbers and *i* satisfies *i*^{2} = −1. The real number *x* is called the **real part** of *z* and is denoted by *x* = Re *z*. The real number *y* is called the **imaginary part** of *z* and is denoted by *y* = Im *z*. We now state several important properties of complex numbers.

**1.** Two complex numbers *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2} are equal if and only if *x*_{1} = *x*_{2} and *y*_{1} = *y*_{2}. In particular, *z* = 0 if and only if Re *z* = 0 and Im *z* = 0.

**2.** Addition of two complex numbers *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2} is defined by

*z*_{1} + *z*_{2} = *x*_{1} + *x*_{2} + *i*(*y*_{1} + *y*_{2}),

and multiplication by a real number *c* is defined by

*cz*_{1} = *cx*_{1} + *icy*_{1}.

Subtraction is then defined by

*z*_{1} − *z*_{2} = *z*_{1} + (−1*z*_{2}) = *x*_{1} − *x*_{2} + *i*(*y*_{1} − *y*_{2}).

**3.** The **complex conjugate** of *z* = *x* + *iy* is the number , that is, the imaginary part of *z* is multiplied by −1. Thus . Note that *z* is real if and only if . Addition and subtraction of a complex number and its complex ...

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