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• Convention for today: <math>x,y,a,b,c,d,...</math> will be real numbers; <math>z,w,u,v,...</math> will be complex numbers |
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• Convention for today: <math>x,y,a,b,c,d,...</math> will be real numbers; <math>z,w,u,v,...</math> will be complex numbers |
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• Dream: Find a field <math>\mathbb C</math> that contains <math>\mathbb R</math> and also contains an element <math>i</math> such that <math>i^2=-1</math>
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Dream: Find a field <math>\mathbb C</math> that contains <math>\mathbb R</math> and also contains an element <math>i</math> such that <math>i^2=-1</math> |
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'''Implications:''' |
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'''Implications:''' |
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• <math>\Rightarrow (a+bi)+(c+di)</math> must be in <math>\mathbb C</math> |
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• <math>\Rightarrow (a+bi)+(c+di)</math> must be in <math>\mathbb C</math> |
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• <math>=(a+c)+(bi+di)</math>
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:<math>=(a+c)+(bi+di)</math> |
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• <math>=(a+c)+(b+d)i</math>
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:<math>=(a+c)+(b+d)i</math> |
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• <math>=e+fi</math>
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:<math>=e+fi</math> |
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• <math>(a+bi)(c+di)=(a+c)+(b+d)i</math>
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<math>(a+bi)(c+di)=(a+c)+(b+d)i</math> |
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• <math>=a(c+di)+bi(c+di)</math>
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:<math>=a(c+di)+bi(c+di)</math> |
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• <math>=ac+adi+bic+bidi</math>
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:<math>=ac+adi+bic+bidi</math> |
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• <math>=ac+bdi^2 + adi+bci</math>
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:<math>=ac+bdi^2 + adi+bci</math> |
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• <math>=(ac-bd)+(ad+bc)i</math>
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:<math>=(ac-bd)+(ad+bc)i</math> |
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• <math>=e+fi</math>
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:<math>=e+fi</math> |
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• <math>0_C=0+0i</math>
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:<math>0_C=0+0i</math> |
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• <math>1_C=1+0i</math>
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:<math>1_C=1+0i</math> |
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• <math>(a+bi)+(c+di)=0+0i</math>
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:<math>(a+bi)+(c+di)=0+0i</math> |
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• <math>-(a+bi)=(-a)+(-b)i</math>
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:<math>-(a+bi)=(-a)+(-b)i</math> |
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• <math>a+bi \neq 0 \Rightarrow (a,b) \neq 0</math>
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:<math>a+bi \neq 0 \Rightarrow (a,b) \neq 0</math> |
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• Find another element of <math>\mathbb C</math>, <math>x+yi</math> such that <math>(a+bi)(x+yi)=(1+0i)</math> |
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• Find another element of <math>\mathbb C</math>, <math>x+yi</math> such that <math>(a+bi)(x+yi)=(1+0i)</math> |
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• <math>(a+bi)(x+yi)=(ax-by)+(ay+bx)i=1+0i</math>
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:<math>(a+bi)(x+yi)=(ax-by)+(ay+bx)i=1+0i</math> |
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• <math>ax-by=1</math> (1)
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:<math>ax-by=1</math> (1) |
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• <math>bx+ay=0</math> (2)
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:<math>bx+ay=0</math> (2) |
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• <math>a,b</math> are given
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:<math>a,b</math> are given |
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• <math>x,y</math> unknowns
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:<math>x,y</math> unknowns |
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• <math>b \times (1)</math> <math>abx-b^2y=b</math> |
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• <math>b \times (1)</math> <math>abx-b^2y=b</math> |
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• <math>a \times (2)</math> <math>abx+a^2y=0</math> |
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• <math>a \times (2)</math> <math>abx+a^2y=0</math> |
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• <math>\Rightarrow a^{2}y+b^{2}y=-b</math>
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:<math>\Rightarrow a^{2}y+b^{2}y=-b</math> |
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• <math>y=\frac{-b}{a^{2}+b^{2}}</math>
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:<math>y=\frac{-b}{a^{2}+b^{2}}</math> |
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• <math>x=\frac{a}{a^{2}+b^{2}}</math>
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:<math>x=\frac{a}{a^{2}+b^{2}}</math> |
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• (Note: We can divide since we assumed that <math>(a,b) \neq 0</math> |
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• (Note: We can divide since we assumed that <math>(a,b) \neq 0</math> |
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• <math>(a+bi)^{-1}=\frac{a}{a^{2}+b^{2}}+\frac{-b}{a^{2}+b^{2}}i=\frac{a-bi}{a^{2}+b^{2}}=\frac{\overline{a+bi}}{|a+bi|^{2}}=\frac{\mbox{conjugate}}{\mbox{norm squared }}</math>
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:<math>(a+bi)^{-1}=\frac{a}{a^{2}+b^{2}}+\frac{-b}{a^{2}+b^{2}}i=\frac{a-bi}{a^{2}+b^{2}}=\frac{\overline{a+bi}}{|a+bi|^{2}}=\frac{\mbox{conjugate}}{\mbox{norm squared }}</math> |
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• Def: Let <math>\mathbb C</math> be the set of all pairs of real numbers <math>{(a,b)}={a+bi}</math>
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Def: Let <math>\mathbb C</math> be the set of all pairs of real numbers <math>{(a,b)}={a+bi}</math> |
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• with <math>+: (a,b)+(c,d)=(a+c,b+d)</math>
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with <math>+: (a,b)+(c,d)=(a+c,b+d)</math> |
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• <math>(a+bi)+(c+di)=(a+c)+(b+d)i</math>
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:<math>(a+bi)+(c+di)=(a+c)+(b+d)i</math> |
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• <math>x:(a+bi)(c+di)=</math>...you know what
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<math>\times :(a+bi)(c+di)=</math>...you know what |
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• 0 = you know what |
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• 0 = you know what |
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• 1 = you know what |
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• 1 = you know what |
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'''Theorem:''' |
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• Thm: |
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• 1. <math>\mathbb C</math> is a field
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#:<math>\mathbb C</math> is a field |
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• 2. <math>(0+1i)^2=(0,1)^2=i^2=-1_{C}=(-1,0)</math>
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#:<math>(0+1i)^2=(0,1)^2=i^2=-1_{C}=(-1,0)</math> |
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• 3. <math>\mathbb R \rightarrow \mathbb C</math> by <math>a \rightarrow a+0i</math>
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#:<math>\mathbb R \rightarrow \mathbb C</math> by <math>a \rightarrow a+0i</math> |
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• Proof: <math>F_{1},F_{2},F_{3},...</math>
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Proof: <math>F_{1},F_{2},F_{3},...</math> |
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• Example: <math>F_{5}</math> (distributivity)
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'''Example:''' <math>F_{5}</math> (distributivity) |
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• Show that <math>z(u+v)=zu+zv</math> |
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• Show that <math>z(u+v)=zu+zv</math> |
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• Let <math>z=(a+bi)</math>
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Let <math>z=(a+bi)</math> |
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• <math>u=(c+di)</math>
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:<math>u=(c+di)</math> |
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• <math>v=(e+fi)</math>
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:<math>v=(e+fi)</math> |
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• When <math>a,b,c,d,e,f \in \mathbb R</math>
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When <math>a,b,c,d,e,f \in \mathbb R</math> |
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• <math>(a+bi)[(c+di)+(e+fi)]=(a+bi)(c+di)+(a+bi)(e+fi)=(ac-bd)+\ldots</math>
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:<math>(a+bi)[(c+di)+(e+fi)]=(a+bi)(c+di)+(a+bi)(e+fi)=(ac-bd)+\ldots</math> |
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• NEXT WEEK: Complex numbers have geometric meaning, geometric interpretation (waves) |
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• NEXT WEEK: Complex numbers have geometric meaning, geometric interpretation (waves) |
Additions to the MAT 240 web site no longer count towards good deed points
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#
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Week of...
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Notes and Links
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1
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Sep 7
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Tue, About, Thu
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2
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Sep 14
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Tue, HW1, HW1 Solution, Thu
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3
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Sep 21
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Tue, HW2, HW2 Solution, Thu, Photo
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4
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Sep 28
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Tue, HW3, HW3 Solution, Thu
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5
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Oct 5
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Tue, HW4, HW4 Solution, Thu,
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6
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Oct 12
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Tue, Thu
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7
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Oct 19
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Tue, HW5, HW5 Solution, Term Test on Thu
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8
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Oct 26
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Tue, Why LinAlg?, HW6, HW6 Solution, Thu
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9
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Nov 2
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Tue, MIT LinAlg, Thu
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10
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Nov 9
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Tue, HW7, HW7 Solution Thu
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11
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Nov 16
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Tue, HW8, HW8 Solution, Thu
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12
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Nov 23
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Tue, HW9, HW9 Solution, Thu
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13
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Nov 30
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Tue, On the final, Thu
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S
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Dec 7
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Office Hours
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F
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Dec 14
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Final on Dec 16
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To Do List
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The Algebra Song!
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Register of Good Deeds
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Misplaced Material
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![09-240-ClassPhoto.jpg](/images/thumb/6/6f/09-240-ClassPhoto.jpg/180px-09-240-ClassPhoto.jpg) Add your name / see who's in!
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NSERC - CMS Math in Moscow Scholarships
The Natural Sciences and Engineering Research Council (NSERC) and the Canadian Mathematical Society (CMS) support scholarships at $9,000 each. Canadian students registered in a mathematics or computer science program are eligible.
The scholarships are to attend a semester at the small elite Moscow Independent University.
Math in Moscow Program http://www.mccme.ru/mathinmoscow/
Application details http://www.cms.math.ca/Scholarships/Moscow
For additional information please see your department or call the CMS at 613-733-2662.
Deadline September 30, 2009 to attend the Winter 2010 semester.
Some links
Class notes for today
• Convention for today:
will be real numbers;
will be complex numbers
Dream: Find a field
that contains
and also contains an element
such that
Implications:
•
•
•
•
must be in
![{\displaystyle =(a+c)+(bi+di)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95c91408913ad69b21407dc8b919cb42af9d05b3)
![{\displaystyle =(a+c)+(b+d)i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c3bbf3c0adc9519f3a2818b7f3c655d3f2b6059)
![{\displaystyle =e+fi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/187bd5e59ea9a72cd3ec844628cd75140565665f)
![{\displaystyle =a(c+di)+bi(c+di)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c1fa70dd5d1af490de0f8b3f130921a5581a5d)
![{\displaystyle =ac+adi+bic+bidi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/211ef2427f85ec9c98bc919d21089c89c17d90d1)
![{\displaystyle =ac+bdi^{2}+adi+bci}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74e22e318f870d9bb083d67906a8c4d94511c0e0)
![{\displaystyle =(ac-bd)+(ad+bc)i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/137a8625abe14cb368a47310a6613146f53fa904)
![{\displaystyle =e+fi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/187bd5e59ea9a72cd3ec844628cd75140565665f)
![{\displaystyle 0_{C}=0+0i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5241ffca4383959229c00ca252bd45f7cb0bf7bd)
![{\displaystyle 1_{C}=1+0i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1a188d9efbe07aba0eeddd9a09fb613f424aef)
![{\displaystyle (a+bi)+(c+di)=0+0i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/183ec08f7a5b75cb62a40579f142c198a72230d1)
![{\displaystyle -(a+bi)=(-a)+(-b)i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f14b9868902d69cb13596df7b47ed88925144f40)
![{\displaystyle a+bi\neq 0\Rightarrow (a,b)\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e50899a07715387e65c3b3226576742f35a6fda)
• Find another element of
,
such that
![{\displaystyle (a+bi)(x+yi)=(ax-by)+(ay+bx)i=1+0i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df83b21f1cce477f778ef140d9f53ad060cbf5f1)
(1)
(2)
are given
unknowns
•
•
![{\displaystyle \Rightarrow a^{2}y+b^{2}y=-b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad7e00b29251957ab52b1b6648a06fd7f108675)
![{\displaystyle y={\frac {-b}{a^{2}+b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f90c4c83c6b7d806055a96e1c83809ae33cfde41)
![{\displaystyle x={\frac {a}{a^{2}+b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21d595e7c759be2a46390cf19aa877cff6953654)
• (Note: We can divide since we assumed that
![{\displaystyle (a+bi)^{-1}={\frac {a}{a^{2}+b^{2}}}+{\frac {-b}{a^{2}+b^{2}}}i={\frac {a-bi}{a^{2}+b^{2}}}={\frac {\overline {a+bi}}{|a+bi|^{2}}}={\frac {\mbox{conjugate}}{\mbox{norm squared }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae95a030030edf8ccd36ed25e979e2b0a8b36a23)
Def: Let
be the set of all pairs of real numbers
with
![{\displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a463db44a55680693e2790a6f351067e9bddbb)
...you know what
• 0 = you know what
• 1 = you know what
Theorem:
is a field
![{\displaystyle (0+1i)^{2}=(0,1)^{2}=i^{2}=-1_{C}=(-1,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28272f9adcb5c36abe6350a45b764d3c0100bbdc)
by ![{\displaystyle a\rightarrow a+0i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fc5b00273611f64f7ed4cacdd3919d1aaae8a8)
Proof:
Example:
(distributivity)
• Show that
Let
![{\displaystyle u=(c+di)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0cbae74cd2f05016538e7371a7eadea772788e4)
![{\displaystyle v=(e+fi)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea3ddd24112e2e17880dc7f3f0f6cd36a66986c7)
When
![{\displaystyle (a+bi)[(c+di)+(e+fi)]=(a+bi)(c+di)+(a+bi)(e+fi)=(ac-bd)+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e197e0c589b3e2d69a1dbf77c74d56e97cf1e9e)
• NEXT WEEK: Complex numbers have geometric meaning, geometric interpretation (waves)