Absolute value in the complex plane

Learning Objectives Define the absolute value (modulus) of a complex number. Calculate the absolute value of a complex number in the form a + bi. Interpret the absolute value of a complex number as its distance from the origin in the complex plane. Calculate the distance between two complex numbers, z₁ and z₂, using the formula |z₁ - z₂|. Apply the properties of the absolute value, such as |z₁z₂| = |z₁||z₂| and |z₁/z₂| = |z₁|/|z₂|, to simplify expressions. Describe and sketch the locus of points in the complex plane defined by equations like |z - c| = r. How do you measure the 'size' or 'magnitude' of a number that exists in two dimensions? 🤔 Let's explore distance in the fascinating world of the complex plane! In this tutorial, you'll discove...

TermDefinitionExample Complex Plane (Argand Diagram)A two-dimensional plane used to represent complex numbers. The horizontal axis is the 'real axis' and the vertical axis is the 'imaginary axis'.The complex number z = 3 + 2i is plotted as the point (3, 2) on the complex plane. Absolute Value (or Modulus)The distance of a complex number from the origin (0, 0) in the complex plane. It is always a non-negative real number.For z = 3 + 4i, the absolute value is |z| = 5, which is the distance from the origin to the point (3, 4). Complex NumberA number of the form z = a + bi, where 'a' is the real part, 'b' is the imaginary part, and i is the imaginary unit (i² = -1).z = -5 + 12i is a complex number with real part -5 and imaginary part 12. Real Part, Re(z...

Definition of Absolute Value (Modulus) For a complex number z = a + bi, its absolute value is |z| = \sqrt{a^2 + b^2} This is the primary formula for calculating the magnitude of a complex number. It is derived directly from the Pythagorean theorem applied to the complex plane, where 'a' and 'b' are the lengths of the two legs of a right triangle and |z| is the hypotenuse. Distance Between Two Complex Numbers The distance between two complex numbers z₁ = a₁ + b₁i and z₂ = a₂ + b₂i is |z₁ - z₂| = \sqrt{(a₁-a₂)^2 + (b₁-b₂)^2} This formula extends the concept of modulus to find the distance between any two points in the complex plane. It is analogous to the distance formula in Cartesian coordinates. Product and Quotient Properties |z₁z₂| = |z₁||z₂| and...