Complex number

Definition

A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with property $i^2 = -1$. Complex numbers can be represented in both rectangular and polar forms.

5 Must Know Facts For Your Next Test

1. The modulus of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$.
2. The polar form of a complex number is expressed as $r(\cos \theta + i\sin \theta)$ or $re^{i\theta}$, where $r = |z|$ and $\theta$ is the argument of the complex number.
3. Addition and subtraction of complex numbers follow the rules $(a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i$.
4. Multiplication of two complex numbers $(a+bi)(c+di)$ results in $(ac-bd) + (ad+bc)i$.
5. The conjugate of a complex number $z = a + bi$ is given by $\overline{z} = a - bi$, useful for dividing complex numbers.

Review Questions

• What is the modulus of the complex number $3 + 4i$?
• Convert the complex number $1 - i$ to its polar form.
• How do you find the product of two complex numbers?

Related terms

Imaginary Unit: $i$, defined such that $i^2 = -1$, used to extend real numbers to complex numbers.

Modulus: The distance from the origin to the point representing a complex number in the complex plane, calculated as $\sqrt{a^2 + b^2}$ for a number $a+bi$.

Argument: The angle $\theta$, measured in radians, between the positive x-axis and the line representing a complex number in polar coordinates.