Transcript
Complex NumbersSolving Quadratics
Complex Numbers012 x
Solving Quadratics
Complex Numbers012 x
Solving Quadratics
solutionsrealno12 x
Complex Numbers
In order to solve this equation we define a new number
012 xSolving Quadratics
solutionsrealno12 x
Complex Numbers
In order to solve this equation we define a new number
012 xSolving Quadratics
solutionsrealno12 x
numberimaginary an is1or 1 2
i ii
Complex Numbers
In order to solve this equation we define a new number
012 xSolving Quadratics
solutionsrealno12 x
numberimaginary an is1or 1 2
i ii
ixx
x
112
073 .. 2 xxge
073 .. 2 xxge
2193
22893
x
073 .. 2 xxge
2193
22893
x
2193 i
073 .. 2 xxge
2193
22893
x
2193 i
2193or
2193 ixix
073 .. 2 xxge
2193
22893
x OR
0732 xx
2193 i
2193or
2193 ixix
073 .. 2 xxge
2193
22893
x OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
073 .. 2 xxge
2193
22893
x OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
04
1923 2
2
ix
073 .. 2 xxge
2193
22893
x OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
04
1923 2
2
ix
0219
23
219
23
ixix
073 .. 2 xxge
2193
22893
x OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
04
1923 2
2
ix
0219
23
219
23
ixix
ixix219
23or
219
23
All complex numbers (z) can be written as; z = x + iy
All complex numbers (z) can be written as; z = x + iyDefinitions;
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 ..
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
(2) If Re(z) = 0, then z is a pure imaginary number
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..
(3) If Im(z) = 0, then z is a real number
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..
(3) If Im(z) = 0, then z is a real number4,,,
43 .. ege
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..
(3) If Im(z) = 0, then z is a real number4,,,
43 .. ege
(4) Every complex number z = x + iy, has a complex conjugate
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..
(3) If Im(z) = 0, then z is a real number4,,,
43 .. ege
(4) Every complex number z = x + iy, has a complex conjugateiyxz
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..
(3) If Im(z) = 0, then z is a real number4,,,
43 .. ege
(4) Every complex number z = x + iy, has a complex conjugateiyxz
izge 72 ..
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
yz
xz
ImRe
izge 53 .. 3Re z 5Im z
(2) If Re(z) = 0, then z is a pure imaginary numberiige 6,3 ..
(3) If Im(z) = 0, then z is a real number4,,,
43 .. ege
(4) Every complex number z = x + iy, has a complex conjugateiyxz
izge 72 .. iz 72
Complex Numbersx + iy
Complex Numbersx + iy
Real Numbersy=0
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions Integers
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
Negatives
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
Negatives
0,0NumbersImaginary
Pure
yx
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
Negatives
0,0NumbersImaginary
Pure
yx
NOTE: Imaginary numbers cannot be ordered
Basic Operations
Basic OperationsAs i a surd, the operations with complex numbers are the same as surds
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication ii 2834
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication ii 2834
2624832 iii
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication ii 2834
2624832 iii 63232 i
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication ii 2834
2624832 iii 63232 i
i3226
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication ii 2834
2624832 iii 63232 i
i3226
Division (Realising The Denominator)
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication ii 2834
2624832 iii 63232 i
i3226
Division (Realising The Denominator)
i
i28
34
ii
2828
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication ii 2834
2624832 iii 63232 i
i3226
Division (Realising The Denominator)
i
i28
34
ii
2828
464624832
ii
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication ii 2834
2624832 iii 63232 i
i3226
Division (Realising The Denominator)
i
i28
34
ii
2828
464624832
ii
i
i
348
3419
681638
Exercise 4A; 1 to 16 evens
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