CHAPTER 1 COMPLEX NUMBERS Dr Nazuhusna Binti Khalid

2 School of Microelectronic 24 February 2021

4 School of Microelectronic 24 February 2021

What you will learn today? STANDARD FORM OPERATIONS THE COMPLEX PLANE THE MODULUS AND ARGUMENT

What is a complex number? It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so. It is defined to be i such that ;

What is a complex number? It is a tool to solve an equation. It has been used to solve equations for the last 200 years or so. It is defined to be i such that ; Or in other words;

Introduction To solve algebraic equations that don’t have the real solutions Real solution No real solution To solve complex numbers : Since :

Powers of i Divide the exponent by 4 No remainder: answer is 1. remainder of 1: answer is i. remainder of 2: answer is – 1. remainder of 3: answer is –i.

Example Solutions:

Introduction Example 1 Simplify: Answer:

Introduction • Definition 1. 1 If z is a complex number, then the standard equation of Complex number denoted by: where a – Real part of z (Re z) b – Imaginary part of z (Im z)

Simplify. -Express these numbers in terms of i. 1. ) 2. ) 3. )

You try… 4. 5. 6.

Introduction Example 1. 2 : Express in the standard form, z: Answer:

Introduction Definition 1. 2 Two complex numbers, z 1 and z 2 are said to be equal if and only if they have the same real and imaginary parts: If and only if a = c and b = d

Introduction Example 1. 3 : Find x and y if z 1 = z 2:

Operations Definition 1. 3 If z 1 = a + bi and z 2 = c + di, then:

Example Solutions:

Operations Example 1. 4 : Given z 1 = 3 -2 i and z 2= 4 -2 i Answer:

Operations Definition 1. 4 The conjugate of z = a + bi can be defined as: **the conjugate of a complex number changes the sign of the imaginary part only!!! **obtained geometrically by reflecting point z on the real axis

Example Solutions:

Operations Example 1. 5 : Find the conjugate of Answer:

The Properties of Conjugate Complex Numbers

Operations Definition 1. 5 (Division of Complex Numbers) If z 1 = a + bi and z 2 = c + di then: Multiply with the conjugate of denominator

Simplify: Multiply the numerator and denominator by the conjugate of 2 + i. In 2 + i, a = 2 and b = 1. a 2 + b 2 = 2 2 + 1 2 – 1 Replace i 2 by – 1 and simplify. Write the answer in the form a + bi. 28 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Operations Example 1. 6 : Simplify and write in standard form, z:

The Complex Plane / Argand Diagram The complex number z = a + bi is plotted as a point with coordinates z(a, b). � Re (z) x – axis � Im (z) y – axis Im(z) b O(0, 0) z(a, b) a Re(z)

The Complex Plane / Argand Diagram Definition 1. 6 (Modulus of Complex Numbers) The modulus of z is defined by Im(z) z(a, b) b r O(0, 0) a Re(z) This definition asserts that the modulus is the distance of the point (a, b) from the origin

The Complex Plane / Argand Diagram Solutions: Exercise: Find the modulus of z:

The Complex Plane / Argand Diagram The Properties of Modulus

Argument of Complex Numbers Definition 1. 7 The argument of the complex number z = a + bi is defined as 2 nd QUADRANT 1 st QUADRANT 3 rd QUADRANT 4 th QUADRANT

Example Solutions:

Argument of Complex Numbers Example 1. 8 : Find the arguments of z:

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