Regions in the Complex Plane

March 7, 2024When finding regions in the complex plane common misconceptions include confusing a number’s modulus with its components and misinterpreting loci inequalities, which can result in inaccurate shading. Recognizing geometric implications, like modulus inequalities forming circles, is key.The tutorials below are designed to help students in identify complex regions within an Argand diagram, complete with summaries and teacher-crafted questions to solidify understanding.Regions in the Complex PlaneContent Criteria:Understanding of the geometric representation of complex numbers.Interpretation and shading of regions based on inequalities involving complex numbers.Familiarity with the modulus of a complex number as a distance in the Argand diagram.Questions for Students:How would you represent the inequality ∣z − 3 − 4i ∣ = 4 on the Argand diagram, and what geometric shape does it correspond to?∣z−5∣>∣z−9∣ does not form a closed region. What does this inequality represent in terms of distance on the Argand diagram?Locus and Combined InequalitiesContent Criteria:Combining inequalities involving the argument of a complex number.Understanding how to represent complex inequalities graphically on an Argand diagram.Identifying the intersection of loci to shade the appropriate region.Questions for Students:What does the inequality 0 < arg ⁡(z + 1 − i) < 3π/π​ represent on the Argand diagram?How would you describe the intersection of the loci described by the given inequalities? Why is the region bounded?Equations and Inequalities Defining LociContent Criteria:Understanding of the locus of a complex number given by an equation ∣z − a − bi∣ = r.Familiarity with loci given by an equality, such as ∣∣z∣ = ∣z − 4i∣, representing the perpendicular bisector of the line segment joining the origin and the point 4i in the Argand diagram.Ability to combine these to find the region that satisfies both conditions.Questions for Students:Draw the locus described by ∣z − 3 − 4i∣ = 5. What does this represent in terms of complex numbers?If ∣z∣ = ∣z − 4i∣, what geometric place does this describe? How does it divide the Argand diagram?Mr Mathematics’ Membership“Unlock a treasure trove of mathematical mastery! 🌟 Join now and gain access to over 800 comprehensive mathematics lessons tailored to propel your teaching to the next level.Why wait? Enhance your curriculum and empower your students.