# 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 + 0i.

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 by 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, SQRT