IB Math HL AHL 1.12 – Complex Numbers
Topic: Number and Algebra
Level: IB Mathematics: Analysis and Approaches (HL)
File Type: Editable Slides / Worksheet
Overview
This resource introduces the concept of complex numbers, expanding the number system beyond the real line to include imaginary and complex values.
Students will learn how to represent, manipulate, and interpret complex numbers both algebraically and geometrically, using technology to support visualization and computation.
The lesson combines algebraic reasoning, graphical interpretation, and applied problem solving to give students a complete understanding of how complex numbers behave and relate to real-world contexts in higher mathematics.
Learning Objectives
By the end of this lesson, students will be able to:
- Identify the real and imaginary parts of a complex number.
- Express complex numbers in Cartesian and polar forms.
- Find the conjugate of a complex number and use it in calculations.
- Perform addition, subtraction, multiplication, and division with complex numbers.
- Illustrate complex numbers and their conjugates on the Argand plane.
- Use technology to solve equations involving complex numbers and visualize their roots.
What’s Included
- Introduction to the imaginary unit (i) and its role in extending the number system.
- Practice problems writing expressions in terms of i with worked solutions.
- Example quadratic equations demonstrating how non-real roots arise from negative discriminants.
- Clear explanation of Cartesian form (a + bi) and polar form r(cosθ + i sinθ) with labeled diagrams.
- Identification of real and imaginary components, including notation for Re(z) and Im(z).
- Definition and examples of the complex conjugate, with geometric interpretation as reflection about the real axis.* Step-by-step practice problems performing arithmetic operations on complex numbers and expressing results in standard form.
- Visual explanation of the Argand plane, showing real and imaginary axes, conjugate symmetry, and plotting examples.
- Use of graphing technology to find and illustrate complex roots of equations.
Topics Covered
- Imaginary numbers and the definition of i.
- Cartesian and polar forms of complex numbers.
- Real and imaginary parts and their notation.
- Complex conjugates and their geometric meaning.
- Operations with complex numbers (addition, multiplication, division).
- Representation on the Argand plane.
- Using technology to solve and visualize complex equations.
Why You’ll Love It
- Connects algebraic manipulation with geometric visualization for deeper understanding.
- Builds essential HL-level foundations for advanced topics such as De Moivre’s theorem and complex transformations.
- Uses clear diagrams and guided examples to reinforce conceptual learning.
- Professionally formatted and classroom-ready with consistent layout and pacing.
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