Topic outline

  • Changes in schools

     This text book is part of the reform of the school curriculum in Rwanda, that is; changes in what is taught in schools and how it is taught. It is hoped that this will make what you learn in school useful to you when you leave school. In the past, the main thing in schooling has been to learn knowledge, that is, facts and ideas about each subject. Now, the main idea is that you should be able to use the knowledge you learn by developing skills or competencies. These skills or competencies include the ability to think for yourself, to be able to communicate with others and explain what you have learnt, and to be creative, that is; developing your own ideas not just following those of the teacher and the text book. You should also be able to find out information and ideas for yourself rather than just relying on what the teacher or text book tells you.

    Activity-based learning 

    This book has a variety of activities for you to do as well as information for you to read. These activities present you with materials or things to do which will help you to learn things and find out things for yourself. You already have a lot of knowledge and ideas based on the experiences you have had and your life within your own community. Some of the activities, therefore, ask you to think about the knowledge and ideas you already have.

    In using this book, therefore, it is essential that you do all the activities. You will not learn properly unless you do these activities. They are the most important part of the book. In some ways, this makes learning more of a challenge. It is more difficult to think for yourself than to copy what the teacher tells you but if you take up this challenge, you will become a better person and become more successful in your life.

    Group work 

    You can learn a lot from other people in your class. If you have a problem, it can often be solved by discussing it with others. Many of the activities in this book, therefore, involve discussion in groups or pairs. Your teacher will help organise these groups and may arrange the classroom so that you are always sitting in groups facing each other. You cannot discuss properly unless you are facing each other.

    Research

     One of the objectives of the competence based curriculum is to help you find things out for yourself. Some activities, therefore, ask you to do research using books in the library, the internet if your school has this, or other sources such as newspapers and magazines. This means that you will develop the skills of learning for yourself when you leave school. Your teacher will help you if your school does not have a good library or internet.

    Icons 

    To guide you, each activity in the book is marked by a symbol or icon to show you what kind of activity it is. The icons are as follows:




    • My Goals

      By the end of this unit, I will be able to:

      ࿌ Identify a real part and imaginary part of a complex

      number.

      ࿌ Convert a complex number from one form to another.

      ࿌ Represent a complex number on Argand diagram.

      ࿌ State De Moivre’s formula and Euler’s formulae.

      ࿌ Apply the properties of complex numbers to perform

      operations on complex numbers in algebraic form, in

      polar form or in exponential form.

      ࿌ Find the modulus and the nth roots of a complex

      number.

      ࿌ Solve in the set of complex numbers a linear or

      quadratic equation.

      ࿌ Use the properties of complex numbers to factorise

      a polynomial and to solve a polynomial equation in

      the set of complex numbers.

      ࿌ Apply complex numbers in trigonometry and

      alternating current problems.

      Introduction 

      Until now, you know several sets of numbers, including  . In each case, the numbers correspond to points on the number line. You already know how to find the square root of a positive real number, but the problem occurs when finding square root of negative numbers.


      1.1. Concepts of complex numbers








      A complex number can be visually represented as a pair of numbers (a ,b) forming a vector from the origin or point on a diagram called Argand diagram (or Argand plane), named after Jean-Robert Argand, representing the complex plane. This plane is also called Gauss plane. The x axis − is called the real axis and is denoted by Re while the y axis − is known as the imaginary axis; denoted Im as illustrated in fig.1.2.
































      Remark

      If b ≠ 0 , from activity 1.11, the sign cannot be taken

      arbitrary because the product xy has sign of b . Then,

      ࿌ If b > 0 , we take the same sign.

      ࿌ If b < 0 , we interchange signs.

      In each case, the complex number has two roots.





      1.2.6. Equations in the set of complex numbers

      Linear equations

      Activity 1.12

      Find the value of z in;

      1. z i + −= 3 40         2.4 43 −+ = − i iz z i

      3. (1 4 + += iiz i )( )   4. (1 2 − =+ iz i )















      An alternative way of defining points in the complex plane, other than using the x and y coordinates, is to use the distance of a point P from the origin together with the angle between the line through P and O and the positive part of the real axis. This idea leads to the polar form of complex numbers. The argument or phase θ (or amplitude) of z is the angle that the radius r makes with the positive real axis, as illustrated in figure 1.5, and is written as arg ( z) .


























      Notice 

      Addition and subtraction in polar form of complex number is not possible directly as it is the case in multiplication and division. For addition and subtraction of complex numbers to be possible, each complex number has to be converted in to Cartesian form first.




























































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