Distance in the complex plane

Learning Objectives Calculate the distance between two complex numbers. Interpret the modulus of a complex number as its distance from the origin. Relate the distance formula in the complex plane to the Pythagorean theorem and the Cartesian distance formula. Write and interpret equations of circles in the complex plane using the distance formula. Describe the locus of points equidistant from two given complex numbers as a perpendicular bisector. Apply the Triangle Inequality to problems involving complex numbers. How is the distance formula you learned in geometry secretly a rule about complex numbers? Let's find out! 🗺️ This tutorial explores how to measure distance in the complex plane. You will learn that the familiar distance formula has a powerful and elegant coun...

TermDefinitionExample Complex Plane (Argand Diagram)A two-dimensional plane used to represent complex numbers. The horizontal axis is the Real Axis (Re) and the vertical axis is the Imaginary Axis (Im).The complex number z = 3 + 4i is plotted as the point (3, 4) in the complex plane. Complex NumberA number of the form z = a + bi, where 'a' is the real part and 'b' is the imaginary part. It can be viewed as a point (a, b) or a vector from the origin to (a, b) in the complex plane.z = -2 + 5i has a real part of -2 and an imaginary part of 5. ModulusThe modulus of a complex number z = a + bi, denoted as |z|, is its distance from the origin in the complex plane.For z = 3 - 4i, the modulus is |z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5. Complex ConjugateThe co...

Modulus of a Complex Number For z = a + bi, the modulus is |z| = \sqrt{a^2 + b^2} This formula calculates the distance of a complex number from the origin. It is a direct application of the Pythagorean theorem on the real and imaginary components. Distance Between Two Complex Numbers The distance between z_1 = a + bi and z_2 = c + di is |z_2 - z_1| = \sqrt{(c-a)^2 + (d-b)^2} To find the distance between two points in the complex plane, first subtract one complex number from the other, then find the modulus of the resulting complex number. The Triangle Inequality |z_1 + z_2| \le |z_1| + |z_2| This states that the length of one side of a triangle (formed by the vectors z_1, z_2, and z_1+z_2) cannot be longer than the sum of the lengths of the other two sides. It provid...