A complex number is an ordered pair of two real numbers (a,b). To represent a complex number, we can use the algebraic notation, z=a+ib with `i^2=-1`. These numerous mathematical resources (calculators, quizzes, games, exercises, course reminders) allow you to practice calculating with complex numbers.

Complex numbers : calculators

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

Complex numbers : games, quizzes and exercises

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

A quadratic equation with real coefficients has in ℂ:
• One real solution if the discriminant Δ=0
• Two real solutions if Δ>0
• Two complex conjugate solutions if and only if Δ<0
For example, the equation `x^2+1=0`, has a negative discriminant, so it admits two complex conjugate solutions.

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