Slide 1 : x2 = 64 Square root of a positive number
Slide 2 : x2 = 64
x = Square root of a positive number
Slide 3 : x2 = 64
x =
x = +8 or -8 Square root of a positive number
Slide 4 : x2 = -64 Square root of a negative number
Slide 5 : x2 = -64
x = + or - Square root of a negative number
Slide 6 : x2 = -64
x = + or -
= Square root of a negative number
Slide 7 : x2 = -64
x = + or -
= = Square root of a negative number
Slide 8 : x2 = -64
x = + or -
= 8 Square root of a negative number
Slide 9 : x2 = -64
x = + or -
= 8
Let = i where i is an imaginary number Square root of a negative number
Slide 10 : x2 = -64
x = + or -
= 8
= 8i
Let = i where i is an imaginary number Square root of a negative number
Slide 11 : Z = A+iB
Complex numbers comprise of a real number A and imaginary number iB Complex Numbers
Slide 12 : Real numbers are represented on a number line. For example +3
-3 -2 -1 0 1 2 3 Complex Numbers on a Number Plane
Slide 13 : Real numbers are represented on a number line. For example -2
-3 -2 -1 0 1 2 3 Complex Numbers on a Number Plane
Slide 14 : Complex numbers are represented on a number plane. One axis is the REAL axis, and the other is the IMAGINARY axis i Complex Numbers on a Number Plane
Slide 15 : Z1 = A+iB
Imaginary Axis (i)
Real Axis Complex Numbers on a Number Plane
Slide 16 : Z1 = A+iB Addition of Complex Numbers
Slide 17 : Z1 = A+iB
Z2 = C+iD Addition of Complex Numbers
Slide 18 : Z1 = A+iB
Z2 = C+iD
Z1 + Z2 = ? Addition of Complex Numbers
Slide 19 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Addition of Complex Numbers
Slide 20 : Z1 = A+iB
Z2 = C+iD
Real Imaginary
Add the Real components A+C
Add the Imaginary components iB +iD Addition of Complex Numbers
Slide 21 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Addition of Complex Numbers Z1 = A+iB
Z2 = C+iD
Z1 + Z2 = (A+C)+i(B+D)
Slide 22 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Addition of Complex Numbers Z1 = 3+i4
Z2 = 7+i6
Z1 + Z2 = (3+7)+i(4+6)
Slide 23 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Addition of Complex Numbers Z1 = 3+i4
Z2 = 7+i6
Z1 + Z2 = (3+7)+i(4+6)
Z1 + Z2 = 10+i10
Slide 24 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Subtraction of Complex Numbers Z1 = A+iB
Z2 = C+iD
Z1 - Z2 = (A-C) + i(B-D)
Slide 25 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Subtraction of Complex Numbers Z1 = 8-i2
Z2 = 5+i9
Z1 - Z2 = (A-C) + i(B-D)
Slide 26 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Subtraction of Complex Numbers Z1 = 8-i2 = 8+i(-2)
Z2 = 5+i9
Z1 - Z2 = (A-C) + i(B-D)
Slide 27 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Subtraction of Complex Numbers Z1 = 8+i(-2)
Z2 = 5+i9
Z1 - Z2 = (8-5) + i(-2-9)
Slide 28 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Subtraction of Complex Numbers Z1 = 8+i(-2)
Z2 = 5+i9
Z1 - Z2 = (8-5) + i(-2-9)
Z1 - Z2 = 3 + i(-11)
Slide 29 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Subtraction of Complex Numbers Z1 = 8+i(-2)
Z2 = 5+i9
Z1 - Z2 = (8-5) + i(-2-9)
Z1 - Z2 = 3 - i11
Slide 30 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Addition and Subtraction of Complex Numbers (3+i9)-(8-i5)+(-2-i3) = ?
Slide 31 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Addition and Subtraction of Complex Numbers (3+i9)-(8-i5)+(-2-i3) =
3 + i9 - 8 + i5 - 2 - i3 = ?
Slide 32 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Addition and Subtraction of Complex Numbers (3+i9)-(8-i5)+(-2-i3) =
3 + i9 - 8 + i5 - 2 - i3 =
3 - 8 - 2 + i9 +i5 - i3 = ?
Slide 33 : Z1 = A+iB
Z2 = C+iD
Real Imaginary Addition and Subtraction of Complex Numbers (3+i9)-(8-i5)+(-2-i3) =
3 + i9 - 8 + i5 - 2 - i3 =
3 - 8 - 2 + i9 +i5 - i3 = -7 +i11
Slide 34 : Z1 = A+iB
Z2 = C+iD
Z1Z2 = ? Multiplication of Complex Numbers
Slide 35 : Z1 = A+iB
Z2 = C+iD
Z1Z2 = (A + iB) (C + iD) Multiplication of Complex Numbers
Slide 36 : Z1 = A+iB
Z2 = C+iD
Z1Z2 = (A + iB) (C + iD)
Z1Z2 = AC + iBC + iAD + i2BD Multiplication of Complex Numbers
Slide 37 : Z1 = A+iB i2 = -1
Z2 = C+iD
Z1Z2 = (A + iB) (C + iD)
Z1Z2 = AC + iBC + iAD + i2BD Multiplication of Complex Numbers
Slide 38 : Z1 = A+iB i2 = -1
Z2 = C+iD
Z1Z2 = (A + iB) (C + iD)
Z1Z2 = AC + iBC + iAD + i2BD
Z1Z2 = AC + iBC + iAD – BD Multiplication of Complex Numbers
Slide 39 : Z1 = A+iB
Z2 = C+iD
Z1Z2 = (A + iB) (C + iD)
Z1Z2 = AC - BD + iBC + iAD Multiplication of Complex Numbers
Slide 40 : Z1 = A+iB
Z2 = C+iD
Z1Z2 = (A + iB) (C + iD)
Z1Z2 = AC - BD + iBC + iAD
Real Imaginary Multiplication of Complex Numbers
Slide 41 : Z1 = 2 + i4
Z2 = 6 - i3
Z1Z2 = (2 + i4) (6 – i3)
Z1Z2 = (2)(6) + (i4)(6) - (i3)(2) –(i3)(i4) Multiplication of Complex Numbers
Slide 42 : Z1 = 2 + i4
Z2 = 6 - i3
Z1Z2 = (2 + i4) (6 – i3)
Z1Z2 = (2)(6) + (i4)(6) - (i3)(2) -(i3)(i4)
Z1Z2 = 12 + i24 - i6 – i212 Multiplication of Complex Numbers
Slide 43 : Z1 = 2 + i4
Z2 = 6 - i3
Z1Z2 = (2 + i4) (6 – i3)
Z1Z2 = 12 + i24 - i6 – i212
i2 = -1 Multiplication of Complex Numbers
Slide 44 : Z1 = 2 + i4
Z2 = 6 - i3
Z1Z2 = (2 + i4) (6 – i3)
Z1Z2 = 12 + i24 - i6 + 12 Multiplication of Complex Numbers
Slide 45 : Z1 = 2 + i4
Z2 = 6 - i3
Z1Z2 = (2 + i4) (6 – i3)
Z1Z2 = 12 + 12 + i24 - i6
Real Imaginary Multiplication of Complex Numbers
Slide 46 : Z1 = 2 + i4
Z2 = 6 - i3
Z1Z2 = (2 + i4) (6 – i3)
Z1Z2 = 24 + i18
Real Imaginary Multiplication of Complex Numbers
Slide 47 : Z1 = A + iB
Z1* = A - iB
Z1* is defined as the complex conjugate Complex Conjugate
Slide 48 : Z1 = A + iB
Z1* = A - iB
Z1 + Z1* = 2A = real number Complex Conjugate
Slide 49 : Z1 = A + iB
Z1* = A - iB
Z1 + Z1* = 2A = real number
Z1 - Z1* = i2B = imaginary number Complex Conjugate
Slide 50 : Z1 = A + iB
Z1* = A - iB
Z1 + Z1* = 2A = real number
Z1 - Z1* = i2B = imaginary number
Z1x Z1* = A2+iAB-iAB-i2B2 Complex Conjugate
Slide 51 : Z1 = A + iB
Z1* = A - iB
Z1 + Z1* = 2A = real number
Z1 - Z1* = i2B = imaginary number
Z1x Z1* = A2-i2B2 Complex Conjugate
Slide 52 : Z1 = A + iB
Z1* = A - iB
Z1 + Z1* = 2A = real number
Z1 - Z1* = i2B = imaginary number
Z1x Z1* = A2+B2 Complex Conjugate
Slide 53 : r = √ (A2+B2) i axis
θ = tan-1 (B/A)
r
θ B
A real axis
Z = A +iB = r cosθ + i rsinθ Polar Form of a Complex Number
Slide 54 : Z1 = 2 + i3
Find the polar form of the complex number Polar Form of a Complex Number
Slide 55 : Z1 = 2 + i3
r = √ (A2+B2)
θ = tan-1 (B/A) Polar Form of a Complex Number
Slide 56 : Z1 = 2 + i3
r = √ (22+32) = √13
θ = tan-1 (3/2) = 56.3o Polar Form of a Complex Number
Slide 57 : Z1 = 2 + i3
r = √ (22+32) = √13
θ = tan-1 (3/2) = 56.3o
Z1 = √13 cos(56.3) + i √13 sin(56.3) Polar Form of a Complex Number
Slide 58 : Z1 = A + iB
r = √ (A2+B2)
θ = tan-1 (B/A)
Z1 = Exponential Form of a Complex Number
Slide 59 : Z1 = 2 + i3
r = √ (22+32) = √13
θ = tan-1 (3/2) = 56.3o
Z1 = √13 ei 56.3 Exponential Form of a Complex Number