x2 t01 01 arithmetic of complex numbers (2013)
DESCRIPTION
TRANSCRIPT
![Page 1: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/1.jpg)
Complex Numbers012 x
Solving Quadratics
![Page 2: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/2.jpg)
Complex Numbers012 x
Solving Quadratics
solutionsrealno12 x
![Page 3: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/3.jpg)
Complex Numbers
In order to solve this equation we define a new number
012 xSolving Quadratics
solutionsrealno12 x
![Page 4: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/4.jpg)
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
![Page 5: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/5.jpg)
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
![Page 6: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/6.jpg)
All complex numbers (z) can be written as; z = x + iy
![Page 7: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/7.jpg)
All complex numbers (z) can be written as; z = x + iyDefinitions;(1) All complex numbers contain a real and an imaginary part
![Page 8: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/8.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
![Page 9: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/9.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 ..
![Page 10: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/10.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
![Page 11: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/11.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 ..
![Page 12: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/12.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
![Page 13: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/13.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 14: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/14.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 .. iz 72
![Page 15: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/15.jpg)
Complex Numbersx + iy
![Page 16: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/16.jpg)
Complex Numbersx + iy
Real Numbersy=0
![Page 17: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/17.jpg)
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
![Page 18: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/18.jpg)
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
![Page 19: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/19.jpg)
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions Integers
![Page 20: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/20.jpg)
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
![Page 21: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/21.jpg)
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
Negatives
![Page 22: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/22.jpg)
Complex Numbersx + iy
Real Numbersy=00
NumbersImaginary y
Rational Numbers
Irrational Numbers
Fractions IntegersNaturals
Zero
Negatives
0,0NumbersImaginary
Pure
yx
![Page 23: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/23.jpg)
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
![Page 24: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/24.jpg)
Basic OperationsAs i a surd, the operations with complex numbers are the same as surds
![Page 25: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/25.jpg)
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
![Page 26: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/26.jpg)
Basic OperationsAs i a surd, the operations with complex numbers are the same as surdsAddition
ii 2834 i 4
Subtraction ii 2834
i512
![Page 27: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/27.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
![Page 28: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/28.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
![Page 29: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/29.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
![Page 30: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/30.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
![Page 31: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/31.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
![Page 32: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/32.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
![Page 33: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/33.jpg)
Conjugate Basics 1 2 1 21 z z z z
![Page 34: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/34.jpg)
Conjugate Basics 1 2 1 21 z z z z
1 2 1 22 z z z z
1 1
2 2
3 z zz z
![Page 35: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/35.jpg)
Conjugate Basics 2 24 zz x y 1 2 1 21 z z z z
1 2 1 22 z z z z
1 1
2 2
3 z zz z
![Page 36: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/36.jpg)
Conjugate Basics 2 24 zz x y 1 2 1 21 z z z z
1 2 1 22 z z z z
1 1
2 2
3 z zz z
15 zz zz
![Page 37: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/37.jpg)
Conjugate Basics 2 24 zz x y 1 2 1 21 z z z z
1 2 1 22 z z z z
1 1
2 2
3 z zz z
15 zz zz
1e.g.
4 3i
![Page 38: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/38.jpg)
Conjugate Basics 2 24 zz x y 1 2 1 21 z z z z
1 2 1 22 z z z z
1 1
2 2
3 z zz z
15 zz zz
1e.g.
4 3i4 316 9
i
4 325 25
i
![Page 39: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/39.jpg)
Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal
ibaibaei 2211If ..
21
21
then
bbaa
![Page 40: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/40.jpg)
Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal
ibaibaei 2211If ..
21
21
then
bbaa
![Page 41: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/41.jpg)
Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal
ibaibaei 2211If ..
. . (i) 2 3 4 2e g i x iy i
21
21
then
bbaa
![Page 42: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/42.jpg)
Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal
ibaibaei 2211If ..
. . (i) 2 3 4 2e g i x iy i
21
21
then
bbaa
2 2 3 3 4 2x iy ix y i
![Page 43: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/43.jpg)
Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal
ibaibaei 2211If ..
. . (i) 2 3 4 2e g i x iy i
21
21
then
bbaa
2 2 3 3 4 2x iy ix y i 2 3 4 and 3 2 2x y x y
![Page 44: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/44.jpg)
Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal
ibaibaei 2211If ..
. . (i) 2 3 4 2e g i x iy i
21
21
then
bbaa
2 2 3 3 4 2x iy ix y i 2 3 4 and 3 2 2x y x y
4 6 8 9 6 6
x yx y
![Page 45: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/45.jpg)
Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal
ibaibaei 2211If ..
. . (i) 2 3 4 2e g i x iy i
21
21
then
bbaa
2 2 3 3 4 2x iy ix y i 2 3 4 and 3 2 2x y x y
4 6 8 9 6 6
x yx y
5 1414 5
x
x
![Page 46: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/46.jpg)
Complex EquationsIf two complex numbers are equal, then their real parts are equal and their imaginary parts are equal
ibaibaei 2211If ..
. . (i) 2 3 4 2e g i x iy i
21
21
then
bbaa
2 2 3 3 4 2x iy ix y i 2 3 4 and 3 2 2x y x y
4 6 8 9 6 6
x yx y
5 1414 5
x
x
14 16 , 5 5
x y
![Page 47: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/47.jpg)
iiwziiwz ii
432 342
![Page 48: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/48.jpg)
iiwziiwz ii
432 342
iiwziiwz
8624342
![Page 49: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/49.jpg)
iiwziiwz ii
432 342
iiwziiwz
8624342
iz
iz
35
32
52 3
![Page 50: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/50.jpg)
iiwziiwz ii
432 342
iiwziiwz
8624342
iz
iz
35
32
52 3
iiwi 34235
32
iiw34
3102
i
iw
35
32
64
610
![Page 51: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/51.jpg)
iiwziiwz ii
432 342
iiwziiwz
8624342
iz
iz
35
32
52 3
iiwi 34235
32
iiw34
3102
i
iw
35
32
64
610
iziw35
32 ,
35
32
![Page 52: X2 t01 01 arithmetic of complex numbers (2013)](https://reader034.vdocuments.us/reader034/viewer/2022051817/5482590b5906b5dd048b4679/html5/thumbnails/52.jpg)
Cambridge: Exercise 1A; 1 to 10 ace etc, 11bd, 13 to 20
Patel: Exercise 4A; 28 ac