operations on complex numbers
z' = `5-i`
Compute `z+z'`.
This type of exercise can be solved using the function : complex_number
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).
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
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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.