Learn to calculate with complex numbers

A selection of free mathematical resources (calculators, exercises, games, quizzes, course reminders) to learn how to calculate with complex numbers.

Complex numbers : Reminder

A complex number is an ordered pair of two real numbers (a, b).

a is called the real part of (a, b);
b is called the imaginary part of (a, b).

To represent a complex number, we use the algebraic notation or algebraic form, z = a + ib with `i^2`=-1

Conjugate of a complex number

The conjugate of a complex number `a+i*b` , where a and b are reals, is the complex number `a-i*b`.

Modulus of a complex number

The modulus of a complex number z=a+ib (where a and b are real) is the positive real number, denoted |z| , defined by : `|z|=sqrt(a^2+b^2)`.

Amplitude of a complex number

The plan has a direct orthogonal reference `(O,vec(i),vec(j))`. Lets z a non zero complex number and M its image. Called the amplitude of the complex number z, any measure, expressed in radians, of the angle `(vec(i),vec(OM))`.

Trigonometric form of a complex number

A complex number z of argument `theta` and modulus r, can be written in its trigonometric form `z=r(cos(theta)+i*sin(theta))`, |z| = r, arg(z) = `theta`.

Exponential notation of a complex number

For any real `theta`, we note `e^(i*theta)` the complex number `cos(theta)+i*sin(theta)`.

A complex number z of amplitude `theta` and modulus r, can be written in its exponential form `z=r*e^(i*theta)`, |z| = r, arg(z) = `theta`.

Second degree equation with real coefficients

One real solution if the discriminant Δ=0
Two real solutions if Δ>0
Two complex conjugate solutions if and only if Δ<0

equation `x^2+1=0`, has a negative discriminant, so it admits two complex conjugate solutions

Complex numbers : calculators

amplitude : amplitude of a complex number. The amplitude calculator determines the amplitude of a complex number from its algebraic form.
complex_conjugate : complex conjugate calculator. The online conjugate calculator returns the conjugate of a complex number.
complex_modulus : complex modulus calculator. The modulus calculator allows you to calculate the modulus of a complex number online.
complex_number : complex number calculator. The complex number calculator allows to perform calculations with complex numbers (calculations with i).
complexe_solve : solving quadratic equation with complex number. The complex number equation calculator returns the complex values for which the quadratic equation is zero.
exp : exponential. The function exp calculates online the exponential of a number.
imaginary_part : imaginary part of complex number. The imaginary part calculator allows you to calculate online the imaginary part of a complex number.
real_part : real part of complex number. The real part calculator allows you to calculate online the real part of a complex number.

Complex numbers : games and quizzes

Quiz on complex numbers (complex numbers): This quiz allows you to apply algebraic calculation techniques to complex numbers.

Complex numbers : exercises

Exercise 1701, algebraic form of a complex number - complex numbers : The goal of this corrected exercise is to write a complex number in its algebraic form z=a+ib.
Exercise 1702, real part of a complex number - complex numbers : To succeed in this exercise, you must know how to determine the real part of a complex expression.
Exercise 1703, imaginary part of a complex number - complex numbers : The purpose of this exercise is to determine with the help of calculation, the imaginary part of a complex number.
Exercise 1704, calculate the conjugate of a complex number - complex numbers : This exercise allows to implement the techniques of calculation of the conjugate of a complex number.
Exercise 1705, operations on complex numbers - complex numbers : The purpose of this exercise is to find the result of arithmetic operations (sum, difference, product) that involve complex numbers.
Exercise 1706, imaginary part of a complex number - complex numbers : The objective of this exercise is to find the imaginary part of a complex number from its algebraic form.
Exercise 1707, real part of a complex number - complex numbers : The objective of this exercise is to find the real part of a complex number from its algebraic form.
Exercise 1708, affix of a complex number - complex numbers : The purpose of this graphing exercise is to place in the plane the affix of a complex number.