Complex Numbers

This topic introduces students to complex numbers and develops their understanding of both the algebraic and geometric ways they can be used to solve mathematical problems. Students learn the arithmetic of complex numbers, including addition, multiplication, division, conjugates, modulus, and arguments, and apply these ideas to solve quadratic and polynomial equations with complex solutions. They explore geometric representations of complex numbers on the complex plane using Cartesian and polar forms, and study how operations such as multiplication correspond to geometric transformations like rotation and dilation. Students also apply important results such as De Moivre’s theorem to find powers and roots of complex numbers, including the nth roots of unity. Through these ideas, they use complex numbers to model lines, circles, and regions on the complex plane and to prove algebraic and geometric results.