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CHAPTER 1

CHAPTER 1. COMPLEX NUMBER. Chapter outline. 1.1 INTRODUCTION AND DEFINITIONS 1.2 OPERATIONS OF COMPLEX NUMBERS 1.3 THE COMPLEX PLANE 1.4 THE MODULUS AND ARGUMENT OF A COMPLEX NUMBER 1.5 THE POLAR FORM OF COMPLEX NUMBERS 1.6 THE EXPONENTIAL FORM OF COMPLEX NUMBERS 1.7 DE MOIVRE`S THEOREM

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CHAPTER 1

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  1. CHAPTER 1 COMPLEX NUMBER

  2. Chapter outline • 1.1 INTRODUCTION AND DEFINITIONS • 1.2 OPERATIONS OF COMPLEX NUMBERS • 1.3 THE COMPLEX PLANE • 1.4 THE MODULUS AND ARGUMENT OF A COMPLEX NUMBER • 1.5 THE POLAR FORM OF COMPLEX NUMBERS • 1.6 THE EXPONENTIAL FORM OF COMPLEX NUMBERS • 1.7 DE MOIVRE`S THEOREM • 1.8 FINDING ROOTS OF A COMPLEX NUMBER • 1.9 EXPANSION FOR COS AND SIN IN TERMS OF COSINES AND SINES • 1.10 LOCI IN THE COMPLEX NUMBER

  3. 1.1 Introduction and Definitions • Complex numbers were discovered in the sixteenth century. • Purpose:- Solving algebraic equations which do not have real solutions. • Complex number, as z, in form of • The number ais real part while b is imaginary part which is combine withjas bj. • where and • By combining the real part and imaginary part, it can solve more quadratic equations.

  4. Example 1.1 Write down the expression of the square roots of • 25 • -25 • Definition 1.1 • If z is a complex number then • Where a is real part and b is imaginary part. Example 1.2 Express in the form i. ii.

  5. Exercise 1.1 : • Simplify • Exercise 1.2: • Express in the form

  6. Definition 1.2 • Two complex numbers are said to be equal if and only if they have the same real and imaginary parts. Example 1.3 Given 5x+2yj = 15 + 4j Exercise 1.3 Given 3x + 7yj = 9 + 28j

  7. 1.2 Operations of Complex Numbers Definition 1.3 If Example 1.4 Given and . Find i. ii. iii. • iv. Determine the value of

  8. Definition 1.4 • The complex conjugate of z = a + bj can defined as Example 1.5 Find the complex conjugate of i. ii. iii. iv.

  9. Exercise 1.4 (complex conjugate): • Find the complex conjugate of

  10. Definition 1.5: Division of Complex Numbers If then Example 1.6 Find the following quantities. Exercise 1.5

  11. 1.3 The Complex Plane • A useful way to visualizing complex numbers is to plot as points in a plane. • The complex number, is plotted as coordinate (a,b). • The x-axis called real axis, y-axis called the imaginary axis. • The Cartesian plane referred as the complex plane or z-plane or Argand diagram.

  12. Example 1.7 Plot the following complex numbers on an Argand diagram. Example 1.8 Given that and that are two complex numbers. Plot in an Argand diagram.

  13. Additional Exercises : • Represent the following complex numbers on an Argand diagram: • Let • a) Plot the complex numbers on an Argand diagram and label them. • b) Plot the complex numbers and on the same Argand diagram.

  14. 1.4 The Modulus and Argument of a Complex Number Definition 1.6 Modulus of Complex Numbers • The norm or modulus or absolute value of z is defined by • Modulus is the distance of the point (a,b) from the origin.

  15. Example 1.9 Find the modulus of the following complex numbers. Exercise 1.6 Find the modulus of the following complex numbers.

  16. Definition 1.7 Argument of Complex Numbers The argument of the complex number, is defined as Example 1.10 Find the arguments of the following complex numbers Exercise 1.7 Find the arguments of the following complex numbers

  17. Additional Exercises: • Find the modulus and argument of complex number below:

  18. 1.5 The Polar of Complex Numbers

  19. Example 1.11 Represent the following complex numbers in polar form. Exercise 1.8 State the following complex numbers in polar form. Example 1.12 Express the following in form. Exercise 1.9

  20. Example 1.13 Given that Find Exercise 1.10 If Find

  21. Additional Exercises: • Write the following numbers in form: • Express the numbers and in the polar form. Find

  22. 1.6 The Exponential form of complex numbers • Definition 1.8 • The exponential form of complex number can be defined as • Where is measured in radians and • Example 1.14 • State the following angles in radians. • Example 1.15 (Exercise 1.12 in Textbook)

  23. Theorem 2 • If and , then • Example 1.16 (Exercise 1.13 in Textbook) • If

  24. Additional Exercises: • Write in exponential form: • 2. Given that • Answer: • 1. 2. • i. i. • ii. Ii.

  25. 1.7 De moivre’s theorem • Theorem 3 • If is a complex number in polar form to any power n, then • with any value n. • Example 1.17(Exercise 1.14 in Textbook) • If

  26. Additional Exercises: • If • Ans: i. 1.732+j ii. -64 • 2. • Ans: • 3. Calculate the • Ans: 32 ab

  27. 1.8 finding roots of a complex number • Theorem 4 • If the n root of z is

  28. Example 1.18(Exercise 1.15 in Textbook): • Find • The square roots of • The cube roots of • Additional Exercises: • Find the square roots of • Ans: i. 5.6568 +5.6568j, -5.6568-5.6568j • ii. 3.5355 +5.5355j, -3.5355-3.5355j

  29. 1.10 Loci in the complex number • Definition 1.9 • A locus in a complex plane is the set of points that have a specified property. A locus of a point in a complex plane could be a straight line, circle, ellipse and etc. • Example 1.20 • If find the equation of the locus defined by:

  30. Additional Exercises: • If , find the equations of the locus defined by:

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