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.