Complex plane

Definition

The complex plane is a two-dimensional space where each point represents a complex number. The horizontal axis is the real part, and the vertical axis is the imaginary part.

5 Must Know Facts For Your Next Test

1. A complex number $z = a + bi$ can be plotted as the point $(a, b)$ on the complex plane.
2. The modulus of a complex number $|z|$ is its distance from the origin, calculated as $|z| = \sqrt{a^2 + b^2}$.
3. The argument (or angle) of a complex number is measured from the positive real axis to the line segment connecting the origin and the point $(a, b)$.
4. Complex conjugates are reflected over the real axis; if $z = a + bi$, then its conjugate is $\overline{z} = a - bi$.
5. Addition and subtraction of complex numbers correspond to vector addition and subtraction in this plane.

Review Questions

• How do you plot a complex number $3 + 4i$ on the complex plane?
• What is the modulus of the complex number $5 - 12i$?
• Explain how to find the argument of a given complex number.

Related terms

Complex Number: A number of form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit satisfying $i^2 = -1$.

Modulus: $|z| = \sqrt{a^2 + b^2}$; represents distance from origin in the complex plane for a given complex number $z = a + bi$.

Argument: \(\text{arg}(z)\); angle formed with positive real axis by line from origin to point representing complex number in polar coordinates.