What is a complex number?
A complex number is written in the form: \( z = a + bi \)
where:
- \( a \) – the real part
- \( b \) – the imaginary part
- \( i \) – the imaginary unit satisfying \( i^2 = -1 \)
Representation in the Complex Plane
The number \( z = a + bi \) is represented as the point \( (a,b) \) in the complex plane.
Vector length (modulus): \( |z| = \sqrt{a^2 + b^2} \)
Angle (argument): \( \theta = \arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \)
Addition of Complex Numbers
If
\( z_1 = a + bi \), \( z_2 = c + di \)
then:
\( z_1 + z_2 = (a + c) + (b + d)i \)
Subtraction of Complex Numbers
\( z_1 - z_2 = (a - c) + (b - d)i \)
Multiplication of Two Complex Numbers
If
\( z_1 = a + bi \), \( z_2 = c + di \)
then:
\( z_1 z_2 = (ac - bd) + (ad + bc)i \)
Multiplication by \( i \)
If
\( z = a + bi \)
then:
\( iz = i(a+bi) = ai + b i^2 = -b + ai \)
That is, multiplying by \( i \) is equivalent to rotating the complex number by \(90^\circ\) counterclockwise in the plane.
Polar Form of a Complex Number
Every complex number can also be written as:
\(z = r(\cos\theta + i\sin\theta) \)
where:
\(r = |z|, \quad \theta = \arg(z)\)
De Moivre's Formula (Powers)
\(z^n = r^n(\cos(n\theta) + i\sin(n\theta))\)
Roots of a Complex Number
The \( n \)-th order roots of a complex number \( z \) are:
\(z_k = \sqrt[n]{r}\left( \cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right) \right)\)
\(k = 0,1,\dots,n-1\)