1. The problem is to understand what complex numbers are and their basic properties. 2. A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit with the property $i^2 = -1$. 3. Important rules: - The real part of the complex number is $a$. - The imaginary part is $b$. - Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules and the fact that $i^2 = -1$. 4. Example: If $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$, then - Addition: $z_1 + z_2 = (3+1) + (4-2)i = 4 + 2i$ - Multiplication: $z_1 \times z_2 = (3)(1) + (3)(-2i) + (4i)(1) + (4i)(-2i) = 3 - 6i + 4i - 8i^2 = 3 - 2i + 8 = 11 - 2i$ 5. Complex numbers are used in many fields such as engineering, physics, and applied mathematics to represent quantities that have both magnitude and direction or phase. Final answer: Complex numbers are numbers of the form $a + bi$ where $a,b \in \mathbb{R}$ and $i^2 = -1$, allowing operations beyond real numbers.