## Description

uA complex number is a combination of real and imaginary numbers. It takes the form, *x + **yi* where x and y are real numbers and I is the imaginary number. The imaginary number has to satisfy the *equation i**2** = 1*. The real numbers correspond to the X axis of a plane while the imaginary numbers correspond to the y-axis. It is used to describe the state of electric circuit elements, describe electromagnetic fields, and to measure two populations.

uA vector is a mathematical operation that has both direction and magnitude. It is represented by a letter. For example, (x,y), or (a,b). Vectors are used to represent any number of physical objects or phenomena. Matrices are often used in the calculation of electrical properties of a circuit and by engineers to calculate mechanical properties of a structure.

uMatrices are ordered arrays numbers in a rectangular form. A matrix looks like this……

uThe complex plane is used to plot complex numbers. Imaginary numbers are plotted on the y axis and real numbers on the X axis.

uFor example: plot the imaginary number −2+4i on the complex plane.

uThe solution is shown in the image alongside. The real values (-2) is plotted on the left side of the X axis. The imaginary values (4) is plotted four units above the x axis as shown by the orange spot…….

uIt corresponds to the form reiθ. In this respect, the real number r is the radial coordinate, and θ is the angular coordinate. The values are defined in terms of Cartesian coordinates; x =rCosθ and y = rSinθ.

uFor example

Sketch the curve represented by the equation r2 = a2 cos 2θ.

uSolution

uCalculate the values as shown in the table

uPlot the values on the polar plane as shown alongside.

uThe curve is symmetrical with respect to the origin, and occurs only with values of θ from -45° to 45° (-¼ π to ¼ π……