Complex Number Formulae

Algebraic Operations with Complex Numbers


Addition:

\[(a+ib) + (c+id) = (a+c) + i(b+d)\]

Subtraction:

\[(a+ib) - (c+id) = (a-c) + i(b-d)\]

Multiplication:

\[(a+ib)(c+id) = (ac-bd) + i(ad+bc)\]

Division:

\[\frac{a+ib}{c+id} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2} \quad \text{(at least one of $c$ and $d$ is non-zero)}\]

Conjugate of a Complex Number


The conjugate of a complex number \(z = a + bi\) is denoted as \(\bar{z}\) and is given by:
\[\bar{z} = a - bi\]

The conjugate of \(z\) has the following properties:

  1. The conjugate of a conjugate is the original complex number: \(\overline{\bar{z}} = z\)

  2. \[\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}\]
    The conjugate of a sum/difference is equal to the sum of the conjugates:
  3. \[\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}\]
    The conjugate of a product is equal to the product of the conjugates:
  4. \[z \cdot \bar{z} = |z|^2 = a^2 + b^2\]
    The conjugate of a complex number multiplied by its conjugate is the real part squared plus the imaginary part squared:
  5. \[z + \bar{z} = 2Re(z)\]
    \[z - \bar{z} = 2iIm(z)\]
    Adding and subtracting a complex number with it’s respective conjugate gives the corresponding real and imaginary parts

Important Properties of Complex Numbers

  1. \(|z| \geq 0\) and \(|z| = 0\) if and only if \(x = 0\) and \(y = 0\) (\(z = 0\)).

  2. \(-|z| \leq \text{Re}(z) \leq |z|\)

  3. \(|z^2| = |z|^2\) or \(|z^n| = |z|^n\), where \(n\) belongs to \(\mathbb{N}\).

  4. \(|z_1z_2 \ldots z_n| = |z_1| \cdot |z_2| \ldots |z_n|\)

  5. \(|z_1 + z_2| \leq |z_1| + |z_2|\)

  6. \(|z_1 - z_2| \geq ||z_1| - |z_2||\)

  7. \(|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)\)

Representation of a Complex Number

Cartesian Representation

A complex number \(z\) in Cartesian form is represented as \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The complex plane is used to visualize Cartesian representations.

Polar Representation

A complex number \(z\) in polar form is represented as \(z = r(\cos \theta + i\sin \theta)\), where \(r\) is the magnitude (modulus) of \(z\), and \(\theta\) is the argument (phase angle) of \(z\). Polar representation is useful for understanding the geometric interpretation of complex numbers. It is also commonly written as \(z = r \angle \theta\).

Conversion from Cartesian to Polar

To convert a complex number \(z = a + bi\) from Cartesian to polar form:
\[r = |z| = \sqrt{a^2 + b^2}\]
\[\theta = \arg(z) = \arctan\left(\frac{b}{a}\right) \quad \text{(in radians)}\]

Conversion from Polar to Cartesian

To convert a complex number \(z = r(\cos \theta + i\sin \theta)\) from polar to Cartesian form:
\[a = r\cos \theta\]
\[b = r\sin \theta\]