Graph complex conjugates

Learning Objectives Define a complex conjugate and identify it for any given complex number. Represent a complex number and its conjugate as coordinate points on the complex plane. Graph a complex number and its conjugate as vectors originating from the origin. By the end of of this lesson, students will be able to articulate the geometric relationship between a complex number and its conjugate as a reflection across the real axis. Prove that the modulus of a complex number is equal to the modulus of its conjugate. Correctly graph the conjugates of purely real and purely imaginary numbers. Ever wondered what a mirror image looks like in the world of numbers? 🤔 Let's explore the perfect reflection of complex numbers on the complex plane! This tutorial will guide you th...

TermDefinitionExample Complex NumberA number that can be expressed in the form z = a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, satisfying i² = -1.z = 4 + 3i Complex Plane (Argand Diagram)A two-dimensional coordinate plane used to represent complex numbers. The horizontal axis is called the 'real axis' (Re) and the vertical axis is called the 'imaginary axis' (Im).The number z = 4 + 3i is plotted at the point (4, 3). Real PartFor a complex number z = a + bi, the real part is the real number 'a'. It corresponds to the x-coordinate on the complex plane.For z = 4 + 3i, the real part is Re(z) = 4. Imaginary PartFor a complex number z = a + bi, the imaginary part is the real number 'b'. It corres...

The Conjugate Rule If z = a + bi, then its conjugate is z̄ = a - bi. This is the fundamental definition. To find the conjugate, you simply change the sign of the imaginary part of the complex number. The real part remains unchanged. The Reflection Property The point (a, -b) representing z̄ is a reflection of the point (a, b) representing z across the real (horizontal) axis. This rule describes the geometric relationship between a complex number and its conjugate on the complex plane. They are mirror images of each other with the real axis acting as the mirror. The Modulus Equality Rule |z| = |z̄| or sqrt(a² + b²) = sqrt(a² + (-b)²) This rule states that a complex number and its conjugate are always the same distance from the origin. This is because squaring the ima...