Absolute values of complex numbers

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 geometrically as its distance from the origin in the complex plane. Apply the properties of absolute values, such as |z₁z₂| = |z₁||z₂|, to simplify expressions. Relate the absolute value of a complex number to its complex conjugate using the formula |z|² = z⋅z̄. Solve basic equations involving the absolute value of complex numbers. How far is the number 5 from zero on a number line? Easy! But how far is the 'number' 3 + 4i from zero in its two-dimensional plane? 🤔 This tutorial explores the absolute value, or modulus, of a complex number, which is its distance from the or...

TermDefinitionExample Complex NumberA number of the form z = a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, satisfying i² = -1.z = 5 - 2i, where a=5 and b=-2. Complex Plane (Argand Diagram)A two-dimensional coordinate plane where the horizontal axis represents the real part (Re) and the vertical axis represents the imaginary part (Im) of a complex number.The complex number 3 + 4i is plotted at the point (3, 4). Absolute Value (or Modulus)The distance of a complex number z = a + bi from the origin (0,0) in the complex plane. It is a non-negative real number, denoted by |z|.For z = 3 + 4i, the absolute value |z| is 5. Real PartThe real number 'a' in a complex number z = a + bi. It is denoted as Re(z).For z = -2 + 7i, Re(z) =...

Definition of Absolute Value For z = a + bi, |z| = \sqrt{a^2 + b^2} This formula is derived directly from the Pythagorean theorem applied to the complex plane, where 'a' and 'b' are the lengths of the legs of a right triangle and |z| is the hypotenuse. Product and Quotient Properties |z_1 z_2| = |z_1| |z_2| and |\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} The absolute value of a product (or quotient) of complex numbers is the product (or quotient) of their individual absolute values. This is often easier than multiplying the numbers first. Relationship with Conjugate |z|^2 = z \cdot \bar{z} The square of the modulus of a complex number is equal to the product of the number and its complex conjugate. This is useful for proofs and for rationalizing de...