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:

Complex number

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 * rboth real and imaginary part of the complex number A are multiplied by the real number r.
A / rboth 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 + rthe real number r is added to the real part of the complex number A.
Arthe real number r is subtracted from the real part of the complex number A.
Addition or subtraction of complex numbers
A + Bthe corresponding parts of the two complex numbers are added.
ABthe 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 / BThe 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

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

TRUNC, ROUND

See also

External links

This site uses cookies to store information on your computer. More info...