Standard form Operations The Cartesian Plane Modulus and Arguments COMPLEX NUMBERS Standard form Operations The Cartesian Plane Modulus and Arguments

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

Introduction In real life, problems usually involve Real Numbers(R). Imaginary number: If we combined Real number and imaginary number: A number that cannot be solved.

Introduction Why do we need to study complex numbers, C ? Many applications especially in engineering: Electrical engineering, Quantum Mechanics and so on. Allow us to solve any polynomial equation, such as:

Introduction To solve algebraic equations that don’t have the real solutions Since, the imaginary number is then Real solution No real solution

Introduction Simplifying a complex number: Since we know that To simplify a higher order of imaginary number:

Introduction Try to simplify Solution

Introduction Simplify (a) (b) (c)

Introduction Definition 1.1 If z is a complex number, then the standard equation of Complex numbers, C denoted by: where a – Real part of z (Re z) b – Imaginary part of z (Im z)

Introduction Example: 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 : Find x and y if z1 = z2:

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

Operations of Complex Numbers Example Given and find:

Operations of Complex Numbers 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 Im(z) Re(z) 3 -3 2 z(2,3)

Operations of Complex Numbers Example : Find the conjugate of:

The Properties of Conjugate Complex Numbers

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

Operations of Complex Numbers Example: Simplify and write in standard form, z:

The Complex Plane/ Cartesian 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 Re(z) O(0,0) a b Im(z) z(a,b)

The Complex Plane/ Cartesian Plane/ Argand Diagram Definition 1.6 (Modulus of Complex Numbers) The modulus of z is defined by *Distance from the origin to z(a,b). r O(0,0) Re(z) a b Im(z) z(a,b)

The Complex Plane/ Cartesian Plane/ Argand Diagram Definition 1.7 The argument of the complex number z = a + bi is defined as arg(z) is not unique. Therefore it can also be written as: 1st QUADRANT 2nd QUADRANT 4th QUADRANT 3rd QUADRANT

The Complex Plane/ Cartesian Plane/ Argand Diagram Example: Find the modulus and the argument of z:

The Complex Plane/ Cartesian Plane/ Argand Diagram The Properties of Modulus

Polar form Exponential Form De Moivre’s Theorem Finding Roots COMPLEX NUMBERS Polar form Exponential Form De Moivre’s Theorem Finding Roots

The Polar Form of Complex Numbers (a,b) r Re(z) Im(z) Based on Figure 1: Applying the Pythagorean trigonometric identity, Therefore, (1)

The Polar Form of Complex Numbers The standard form of complex numbers is given by: Definition: Then the polar form is defined by:

The Polar Form of Complex Numbers Example: Represent the following complex numbers in polar form: a) b) c)

The Polar Form of Complex Numbers Example: Express the following in the standard form complex numbers: a) b) c)

The Polar Form of Complex Numbers Theorem 1: If z1 and z2 are complex numbers in polar form where, Then, a) b)

The Polar Form of Complex Numbers Example: Find and , if: a) b)

The Exponential Form of Complex Numbers Euler’s formula state that for any real number Where is the exponential function, i is the imaginary unit, sine and cosines are trigonometric function and arg (z) = is in radians. Definition: The exponential form of a complex numbers can be defined as:

The Exponential Form of Complex Numbers Example: Represent the following complex numbers in exponential form: a) b) c)

The Exponential Form of Complex Numbers Example: Express the following in the standard form complex numbers: a) b)

The Exponential Form of Complex Numbers Theorem 2: If z1 and z2 are complex numbers in exponential form where, Then, a) b)

The Exponential Form of Complex Numbers Example: Find and , if: a) b)

De Moivre’s Theorem Let z1 and z2 be complex numbers where Then: From the properties of polar form:

De Moivre’s Theorem From the properties of modulus: And suppose:

De Moivre’s Theorem Using all these facts; (3),(4) and (5), we can compute the square of a complex number. Suppose so Then

De Moivre’s Theorem Theorem 3: If is a complex number in polar form to any power of n, then De Moivre’s Theorem: Therefore :

De Moivre’s Theorem Example: a)Let Find b) Use De Moivre’s theorem to find: (i) (ii)

De Moivre’s Theorem : Finding Roots We know that argument of z is not unique, then we can also defined Using the fact above and DMT, we can find the roots of a complex number,

De Moivre’s Theorem : Finding Roots Theorem 4 If then, the n root of z is: (θ in degrees) OR (θ in radians) Where k = 0,1,2,..n-1

De Moivre’s Theorem : Finding Roots Example Find all complex cube roots of

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De Moivre’s Theorem : Finding Roots Sketch on the complex plane: 1 y x nth roots of unity: Roots lie on the circle with radius 1

De Moivre’s Theorem : Finding Roots Example: Solve and show the roots on the Argand diagram.

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De Moivre’s Theorem : Finding Roots Sketch on complex plane: y x

COMPLEX NUMBERS Expansions for cosn and sinn in terms of Cosines and Sines of multiple n Loci in the Complex Numbers

Expansion of Sin and Cosine Theorem 5: If , then: Theorem 6: (Binomial Theorem) If , then:

Expansion of Sin and Cosine Example Expand using binomial theorem, then write in standard form of complex number:

Expansion of Sin and Cosine Example State in terms of cosines.

Expansion of Sin and Cosine Example: By using De Moivre’s theorem and Binomial theorem, prove that:

Expansion of Sin and Cosine Example Using appropriate theorems, state the following in terms of sine and cosine of multiple angles :

Loci in the Complex Numbers Since any complex number, z = x+iy correspond to point (x,y) in complex plane, there are many kinds of regions and geometric figures in this plane can be represented by complex equations or inequations. Definition 1.9 A locus in a complex plane is the set of points that have specified property. A locus in a complex plane could be a straight line, circle, ellipse and etc.

Loci in the Complex Numbers (i) Standard form of circle equation Equation of circle with center at the origin, z0 , O(0,0) and z=x+iy, P(x,y) and radius, r y x P on circumference: P outside circle: P inside circle:

Loci in the Complex Numbers (i) Standard form of circle equation Equation of circle with center, z1= x1+ iy1, A(x1, y1) and z=x+iy, P(x,y) and radius, r

Loci in the Complex Numbers Example: What is the equation of circle in complex plane with radius 2 and center at 1+i Solution: Re Im Distance from center to any point P must be the same

Loci in the Complex Numbers (ii) Perpendicular bisector where the distance from point z=x+iy, P(x,y) to z1 =x1+iy1 , A(x1,y1 ) and z2 = x2+iy2, B(x2,y2 ) are equal. Locus of z represented by perpendicular bisector of line segment joining the points to z1 and z2

Loci in the Complex Numbers Example Find the equation of locus if:

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Loci in the Complex Numbers Re Im Distance from point (0,-1) and (2,0) to any point P must be the same

Loci in the Complex Numbers (ii) Argument: The locus of z represented by a half line from the fixed point z1 =x1+iy1 , making an angle, with a line from the fixed point z1 which is parallel to x-axis

Loci in the Complex Numbers Example If , determine the equation of loci and describe the locus of z

Loci in the Complex Numbers Example Find the equation of locus if: a) b) c)