Find the modulus and argument of a complex number

Learning Objectives Define the modulus and argument of a complex number in the context of an Argand diagram. Calculate the modulus of any complex number z = a + bi using the formula |z| = sqrt(a^2 + b^2). Determine the correct quadrant for a complex number based on the signs of its real and imaginary parts. Calculate the principal argument (Arg z) of a complex number, making necessary adjustments for quadrants II, III, and IV. Find the modulus and argument for complex numbers lying on the real or imaginary axes. Express a complex number's modulus and argument with precision, using both exact values (in terms of pi) and decimal approximations. How can we describe a point's location not by its (x, y) coordinates, but by its distance and direction from a starting poin...

TermDefinitionExample Complex Number (Rectangular Form)A number of the form z = a + bi, where 'a' is the real part and 'b' is the imaginary part. It can be plotted as a point (a, b) on a complex plane.z = 3 + 4i has a real part a = 3 and an imaginary part b = 4. Argand DiagramA two-dimensional plane used to plot complex numbers. The horizontal axis is the real axis (Re), and the vertical axis is the imaginary axis (Im).The number z = -2 + i is plotted at the point (-2, 1) on the Argand diagram. ModulusThe distance of the complex number from the origin (0,0) on the Argand diagram. It is a non-negative real number, denoted as |z| or r.For z = 3 + 4i, the modulus |z| is 5. This means the point (3, 4) is 5 units away from the origin. ArgumentThe angle θ (theta), measured i...

Formula for the Modulus (r) For a complex number z = a + bi, the modulus is: |z| = r = \sqrt{a^2 + b^2} This is essentially the Pythagorean theorem applied to the Argand diagram. Square the real part and the imaginary part, add them together, and take the square root. Note that you use 'b', not 'bi'. Formula for the Principal Argument (θ) For z = a + bi, let the reference angle be \alpha = \arctan\left(\frac{|b|}{|a|}\right). Then the Principal Argument \theta is found by quadrant: • Q1 (a>0, b>0): \theta = \alpha • Q2 (a<0, b>0): \theta = \pi - \alpha • Q3 (a<0, b<0): \theta = -(\pi - \alpha) = \alpha - \pi • Q4 (a>0, b<0): \theta = -\alpha First, calculate the reference angle α using the absolute values of a and b. Then, use the...