Complex plane

Definition

The complex plane is a two-dimensional plane used to represent complex numbers graphically. Each point on the plane corresponds to a unique complex number, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.

5 Must Know Facts For Your Next Test

1. A complex number $z = a + bi$ can be represented as the point $(a,b)$ on the complex plane.
2. The modulus of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$, representing its distance from the origin.
3. The argument (or angle) of a complex number, denoted as $\arg(z)$, is the angle formed with the positive real axis and can be calculated using $\theta = \tan^{-1}(\frac{b}{a})$.
4. Complex numbers can also be represented in polar form as $z = r(\cos \theta + i \sin \theta)$ or $z = re^{i\theta}$ where $r$ is the modulus and $\theta$ is the argument.
5. Addition and subtraction of complex numbers correspond to vector addition and subtraction in the complex plane.

Review Questions

• How do you represent a complex number like $3 + 4i$ on the complex plane?
• What are the modulus and argument of the complex number $1 - i$?
• Convert the polar form representation of a complex number back to its rectangular form.

Related terms

Modulus: The modulus (or absolute value) of a complex number $z = a + bi$, denoted by $|z|$, is given by $\sqrt{a^2 + b^2}$.

Argument: The argument of a complex number, denoted as $\arg(z)$, is the angle that the line representing the complex number makes with the positive real axis in radians.

Polar Form: $Polar \, form \, represents \,a \,complex \,number \,as \, z = r(\cos \theta + i \sin \theta)$ or equivalently as $re^{i\theta}$ where $$r$$ is $$the$$ modulus and $$θ$$ is $$the$$ argument.