x2 t01 01 definitions (2010)
TRANSCRIPT
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Complex NumbersSolving Quadratics
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Complex Numbers012 x
Solving Quadratics
![Page 3: X2 T01 01 definitions (2010)](https://reader033.vdocuments.site/reader033/viewer/2022042819/55cc5dadbb61eb95138b45cb/html5/thumbnails/3.jpg)
Complex Numbers012 x
Solving Quadratics
solutionsrealno12 x
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Complex Numbers
In order to solve this equation we define a new number
012 xSolving Quadratics
solutionsrealno12 x
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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
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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
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073 .. 2 xxge
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073 .. 2 xxge
2193
22893
x
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073 .. 2 xxge
2193
22893
x
2193 i
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073 .. 2 xxge
2193
22893
x
2193 i
2193or
2193 ixix
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073 .. 2 xxge
2193
22893
x OR
0732 xx
2193 i
2193or
2193 ixix
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073 .. 2 xxge
2193
22893
x OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
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073 .. 2 xxge
2193
22893
x OR
0732 xx
2193 i
2193or
2193 ixix
04
1923 2
x
04
1923 2
2
ix
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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
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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
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All complex numbers (z) can be written as; z = x + iy
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All complex numbers (z) can be written as; z = x + iyDefinitions;
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All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
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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
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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 ..
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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
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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
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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
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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 ..
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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
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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
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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
![Page 28: X2 T01 01 definitions (2010)](https://reader033.vdocuments.site/reader033/viewer/2022042819/55cc5dadbb61eb95138b45cb/html5/thumbnails/28.jpg)
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
![Page 29: X2 T01 01 definitions (2010)](https://reader033.vdocuments.site/reader033/viewer/2022042819/55cc5dadbb61eb95138b45cb/html5/thumbnails/29.jpg)
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 ..
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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
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Complex Numbersx + iy
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Complex Numbersx + iy
Real Numbersy=0
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Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
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Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
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Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
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Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions
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Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions Integers
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Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
![Page 39: X2 T01 01 definitions (2010)](https://reader033.vdocuments.site/reader033/viewer/2022042819/55cc5dadbb61eb95138b45cb/html5/thumbnails/39.jpg)
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
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Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
Negatives
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Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
Negatives
0,0NumbersImaginary
Pure
yx
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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
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Basic Operations
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Basic OperationsAs i a surd, the operations with complex numbers are the same as surds
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Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
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Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834
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Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
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Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction
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Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
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Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
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Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
Multiplication
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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
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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
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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
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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
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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)
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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
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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
![Page 59: X2 T01 01 definitions (2010)](https://reader033.vdocuments.site/reader033/viewer/2022042819/55cc5dadbb61eb95138b45cb/html5/thumbnails/59.jpg)
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
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Exercise 4A; 1 to 16 evens