Complex numbers (other)

Beyond basic arithmetic, complex numbers can be represented in different forms, such as polar form, and can be used to solve equations that have no real solutions. The polar form of a complex number uses its distance from the origin (modulus) and the angle it makes with the positive real axis (argument).

Important properties

• A complex number $z = a + bi$ can also be written as $z = r( ext{cos} heta + i ext{sin} heta)$, where $r = |z|$ is the modulus and $\theta$ is the argument.

• Multiplying two complex numbers in polar form multiplies their moduli and adds their arguments.

• The conjugate of $z = a + bi$ is $\overline{z} = a - bi$.

• The modulus $|z| = \sqrt{a^2 + b^2}$ gives the distance from the origin in the complex plane.