# Complex numbers

A complex number is a number that can be expressed in the form *a* + *bi* where *a* and *b* are real numbers and *i* is the imaginary unit, satisfying the equation *i*^{ 2}=−1. The number *a* is called the real part and the number *b* is called the imaginary part of the complex number.

Any real number *r* can be expressed as a complex number *r* + 0*i*.

The complex number *a* + *bi* can be represented visually as follows:

The phase and norm (modulus) are alternative ways to define a complex number.

## Create a complex number

You create a complex number using the COMPLEX function or by entering the number directly in the form a+bi. COMPLEX(a,b) returns the number *a* + *bi*. For example, COMPLEX(2,3) returns 2+3i.

## Basic mathematical operations

Multiplication or division of a complex number with a real number | |

A * r | both real and imaginary part of the complex number A are multiplied by the real number r. |

A / r | both real and imaginary part of the complex number A divided by the real number r. |

Addition or subtraction of a complex number with a real number | |

A + r | the real number r is added to the real part of the complex number A. |

A − r | the real number r is subtracted from the real part of the complex number A. |

Addition or subtraction of complex numbers | |

A + B | the corresponding parts of the two complex numbers are added. |

A − B | the corresponding parts of the two complex numbers are subtracted. |

Multiplication of complex numbers | |

A * B | The multiplication of two complex numbers is defined as follows: $$ (a+bi) (c+di) = (ac-bd) + (bc+ad)i $$ |

Division of complex numbers | |

A / B | The division of two complex numbers is defined as follows: $$ \frac{a + bi}{c + di} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i $$ |

## Complex number functions

- COMPLEX(
*a*,*b*) Create complex number - RE(
*A*) Returns the real part of a complex number - IM(
*A*) Returns the imaginary part of a complex number - NORM(
*A*) Returns the norm (or modulus) of a complex number - CONJ(
*A*) Returns the complex conjugate of a complex number - PHASE(
*A*) Returns the phase angle (or angular component) of a complex number - POLAR(
*rho*,*theta*) Returns the complex number defined by its polar components - CRND() Returns a random complex number.
- CSQRT(
*x*) Square root function.

## Mathematical functions on complex numbers

The following mathematical functions accept a complex number as argument.

ABS, EXP, INV, LN, LOG, POWER, POW10, SQRT, SQUARE

ACOS, ACOSH, ASIN, ASINH, ATAN, ATANH, COS, COSH, SIN, SINH, TAN, TANH