Introduction to the complex plane

Learning Objectives Represent a complex number z = a + bi as a point (a, b) in the complex plane. Identify and label the real and imaginary axes. Calculate the modulus (or absolute value) of a complex number and interpret it as the distance from the origin. Plot the complex conjugate of a number and describe its geometric relationship to the original number. Graphically represent the sum and difference of two complex numbers using vector addition (the parallelogram law). Calculate the distance between two complex numbers in the plane. We can plot real numbers on a one-dimensional line, but where in the world do we plot a number like 3 + 4i? 🤔 This tutorial introduces the complex plane, a two-dimensional grid that extends the number line to visualize complex numbers. You wi...

TermDefinitionExample Complex Plane (or Argand Diagram)A two-dimensional coordinate plane used to represent complex numbers. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.The number z = 2 + 5i is represented by the point (2, 5) on the complex plane. Real AxisThe horizontal axis of the complex plane. It represents all numbers where the imaginary part is zero (e.g., -3, 0, 5.5).The number z = 4 is plotted at the point (4, 0) on the real axis. Imaginary AxisThe vertical axis of the complex plane. It represents all numbers where the real part is zero (e.g., -2i, i, 3i).The number z = -2i is plotted at the point (0, -2) on the imaginary axis. ModulusThe modulus of a complex number z = a + bi, denoted |z|, is its distance from the origin (0,0) in the complex...

Plotting a Complex Number z = a + bi \rightarrow (a, b) To plot a complex number, identify its real part 'a' and its imaginary part 'b'. The real part is the x-coordinate, and the imaginary part is the y-coordinate on the complex plane. Modulus Formula |z| = |a + bi| = \sqrt{a^2 + b^2} This formula is derived from the Pythagorean theorem, where 'a' and 'b' are the legs of a right triangle and |z| is the hypotenuse. Distance Between Two Complex Numbers d = |z_1 - z_2| = \sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2} The distance between two complex numbers, z_1 and z_2, is the modulus of their difference. This is identical to the distance formula in a standard Cartesian plane.