CHAPTER 1 COMPLEX NUMBERS STANDARD FORM OPERATIONS THE COMPLEX PLANE THE MODULUS AND ARGUMENT THE POLAR FORM

Classification of Numbers INTEGERS (Z) COMPLEX NUMBERS (C) REAL NUMBERS (R) RATIONAL NUMBERS (Q) IRRATIONAL NUMBERS WHOLE NUMBERS (W) NATURAL NUMBERS (N)

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

Introduction Example 1 Simplify:

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)

Introduction Example 1.2 : Express in the standard form, z:

Introduction Definition 1.2 2 complex numbers, z1 and z2 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 z1 = z2:

Operations Definition 1.3 If z1 = a + bi and z2 = c + di, then:

Operations Example 1.4 : Given z1 = 3-2i and z2= 4-2i

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

Operations Example 1.5 : Find the conjugate of

The Properties of Conjugate Complex Numbers

Operations Definition 1.5 (Division of Complex Numbers) If z1 = a + bi and z2 = c + di then: Multiply with the conjugate of denominator

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) Re(z) O(0,0) z(a,b) a b

The Complex Plane / Argand Diagram Definition 1.6 (Modulus of Complex Numbers) The modulus of z is defined by Im(z) Re(z) O(0,0) z(a,b) a b r

The Complex Plane / Argand Diagram Definition 1.6 (Modulus of Complex Numbers) The modulus of z is defined by Im(z) Re(z) O(0,0) z(a,b) a b r

The Complex Plane / Argand Diagram Example 1.7 : 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 1st QUADRANT 2nd QUADRANT 4th QUADRANT 3rd QUADRANT

Argument of Complex Numbers Example 1.8 : Find the arguments of z: