Absolute values of complex numbers

Learning Objectives Define the absolute value (or modulus) of a complex number. Calculate the absolute value of a complex number in the form a + bi using the distance formula. Interpret the absolute value of a complex number as its distance from the origin on the complex plane. Apply the properties of absolute values to products and quotients of complex numbers. By the end of a this lesson, students will be able to compare the magnitudes of two or more complex numbers. Solve simple equations involving the absolute value of a complex number. If a real number's absolute value is its distance from zero on a number line, how do we measure the 'distance' of a number that exists in two dimensions? 🗺️ This tutorial explores the concept of the absolute value, or modu...

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 = 5 - 2i is a complex number where a = 5 and b = -2. Complex Plane (Argand Diagram)A two-dimensional coordinate plane where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part.The complex number z = 3 + 4i is plotted at the point (3, 4) on the complex plane. Absolute Value (or Modulus)The absolute value of a complex number z = a + bi, denoted as |z|, is its distance from the origin (0, 0) on the complex plane.For z = 3 + 4i, its absolute value |z| is 5, because the point (3, 4) is 5 units away from the origin. Real PartF...

Formula for Absolute Value (Modulus) For a complex number z = a + bi, its absolute value is |z| = \sqrt{a^2 + b^2} This formula is derived from the Pythagorean theorem on the complex plane. Use it to calculate the magnitude of any complex number by identifying its real part 'a' and imaginary part 'b'. Product Property |z_1 \cdot z_2| = |z_1| \cdot |z_2| The absolute value of the product of two complex numbers is equal to the product of their individual absolute values. This is often easier than multiplying the complex numbers first. Quotient Property |\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}, \text{ where } z_2 \neq 0 The absolute value of the quotient of two complex numbers is the quotient of their individual absolute values. This avoids the need f...