Drorbn - User contributions [en]

14-240/Classnotes for Monday November 10

2014-11-19T01:55:51Z

Yue.Jiang:

The pdf notes for Wednesday class is: {{14-240/Navigation}}

By Yue Jiang--[[User:Yue.Jiang|Yue.Jiang]] ([[User talk:Yue.Jiang|talk]]) 11:29, 22 October 2014 (EDT)

----

<gallery>
File:monday Nov 10.jpg|Nov 10 note

14-240/Classnotes for Monday November 10

2014-11-19T01:55:38Z

Yue.Jiang: Created page with "The pdf notes for Wednesday class is: {{14-240/Navigation}} By Yue Jiang--Yue.Jiang (talk) 11:29, 22 October 2014 (EDT) ---- <gal..."

14-240/Navigation

2014-11-19T01:54:45Z

Yue.Jiang:

<noinclude>Back to [[14-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center style="color: red;"
|colspan=3|'''Welcome to Math 240!''' 
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-240/About This Class|About This Class]], What is this class about? ({{Pensieve link|Classes/14-240/one/What_is_This_Class_AboutQ.pdf|PDF}}, {{Pensieve link|Classes/14-240/What_is_This_Class_AboutQ.html|HTML}}), [[14-240/Classnotes for Monday September 8|Monday]], [[14-240/Classnotes for Wednesday September 10|Wednesday]]
|- align=left
|align=center|2
|Sep 15
|[[14-240/Homework Assignment 1|HW1]], [[14-240/Classnotes for Monday September 15|Monday]], [[14-240/Classnotes for Wednesday September 17|Wednesday]], {{Pensieve link|Classes/14-240/nb/TheComplexField.pdf|TheComplexField.pdf}} 
|- align=left
|align=center|3
|Sep 22
|[[14-240/Homework Assignment 2|HW2]], [[14-240/Class Photo|Class Photo]], [[14-240/Classnotes for Monday September 22|Monday]], [[14-240/Classnotes for Wednesday September 24|Wednesday]] 
|- align=left
|align=center|4
|Sep 29
|[[14-240/Homework Assignment 3|HW3]], [[14-240/Classnotes for Wednesday October 1|Wednesday]], [[14-240/Tutorial-Sep30|Tutorial]] 
|- align=left
|align=center|5
|Oct 6
|[[14-240/Homework Assignment 4|HW4]], [[14-240/Classnotes for Monday October 6|Monday]], [[14-240/Classnotes for Wednesday October 8|Wednesday]], [[14-240/Tutorial-October7|Tutorial]] 
|- align=left
|align=center|6
|Oct 13
|No Monday class (Thanksgiving), [[14-240/Classnotes for Wednesday October 15|Wednesday]], [[14-240/Tutorial-October14|Tutorial]]
|- align=left
|align=center|7
|Oct 20
|[[14-240/Homework Assignment 5|HW5]], [[14-240/Term Test|Term Test]] at tutorials on Tuesday, [[14-240/Classnotes for Wednesday October 22|Wednesday]]
|- align=left
|align=center|8
|Oct 27
|[[14-240/Homework Assignment 6|HW6]], [[14-240/Classnotes for Monday October 27|Monday]], [[14-240/Linear Algebra - Why We Care|Why LinAlg?]], [[14-240/Classnotes for Wednesday October 29|Wednesday]], [[14-240/Tutorial-October28|Tutorial]]
|- align=left
|align=center|9
|Nov 3
|Monday is the last day to drop this class, [[14-240/Homework Assignment 7|HW7]], [[14-240/Classnotes for Monday November 3|Monday]], [[14-240/Classnotes for Wednesday November 5|Wednesday]]
|- align=left
|align=center|10
|Nov 10
|[[14-240/Homework Assignment 8|HW8]], [[14-240/Classnotes for Monday November 10|Monday]]
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT November break
|- align=left
|align=center|12
|Nov 24
|[[14-240/Homework Assignment 9|HW9]]
|- align=left
|align=center|13
|Dec 1
|Wednesday is a "makeup Monday"!
|- align=left
|align=center|F
|Dec 8
|[[14-240/The Final Exam|Our Final Exam]] will take place on Wednesday December 10, 2-5PM, at GB 404 for students with last names between A and Lo, and at GB 412 for the rest.
|- align=left
|colspan=3 align=center|[[14-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-240-ClassPhoto.jpg|310px|Class Photo]] [[14-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:14-240-Splash.png|310px]]
|}

14-240/Classnotes for Wednesday November 5

2014-11-19T01:53:51Z

Yue.Jiang:

The pdf notes for Wednesday class are: {{14-240/Navigation}}

By Yue Jiang--[[User:Yue.Jiang|Yue.Jiang]] ([[User talk:Yue.Jiang|talk]]) 11:29, 22 October 2014 (EDT)

----

<gallery>
File:wednesday1.jpg|Nov 5 note1

File:wednesday2.jpg|Nov 5 note2

File:wednesday3.jpg|Nov 5 note3

14-240/Classnotes for Wednesday November 5

2014-11-19T01:53:22Z

Yue.Jiang: Created page with "The pdf notes for Wednesday class is: {{14-240/Navigation}} By Yue Jiang--Yue.Jiang (talk) 11:29, 22 October 2014 (EDT) ---- <gal..."

File:Monday Nov 10.jpg

2014-11-19T01:51:47Z

Yue.Jiang:

File:Wednesday3.jpg

2014-11-19T01:51:12Z

Yue.Jiang:

File:Wednesday2.jpg

2014-11-19T01:50:55Z

Yue.Jiang:

File:Wednesday1.jpg

2014-11-19T01:50:28Z

Yue.Jiang:

File:Monday.jpg

2014-11-19T01:50:00Z

Yue.Jiang:

14-240/Classnotes for Monday November 3

2014-11-19T01:49:27Z

Yue.Jiang: Created page with "The pdf notes for Monday class: {{14-240/Navigation}} By Yue Jiang--Yue.Jiang (talk) 11:29, 22 October 2014 (EDT) ---- <gallery> ..."

The pdf notes for Monday class: {{14-240/Navigation}}

By Yue Jiang--[[User:Yue.Jiang|Yue.Jiang]] ([[User talk:Yue.Jiang|talk]]) 11:29, 22 October 2014 (EDT)

----

<gallery>
File:monday.jpg|Nov 3 note

14-240/Navigation

2014-11-19T01:47:47Z

Yue.Jiang:

<noinclude>Back to [[14-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center style="color: red;"
|colspan=3|'''Welcome to Math 240!''' 
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-240/About This Class|About This Class]], What is this class about? ({{Pensieve link|Classes/14-240/one/What_is_This_Class_AboutQ.pdf|PDF}}, {{Pensieve link|Classes/14-240/What_is_This_Class_AboutQ.html|HTML}}), [[14-240/Classnotes for Monday September 8|Monday]], [[14-240/Classnotes for Wednesday September 10|Wednesday]]
|- align=left
|align=center|2
|Sep 15
|[[14-240/Homework Assignment 1|HW1]], [[14-240/Classnotes for Monday September 15|Monday]], [[14-240/Classnotes for Wednesday September 17|Wednesday]], {{Pensieve link|Classes/14-240/nb/TheComplexField.pdf|TheComplexField.pdf}} 
|- align=left
|align=center|3
|Sep 22
|[[14-240/Homework Assignment 2|HW2]], [[14-240/Class Photo|Class Photo]], [[14-240/Classnotes for Monday September 22|Monday]], [[14-240/Classnotes for Wednesday September 24|Wednesday]] 
|- align=left
|align=center|4
|Sep 29
|[[14-240/Homework Assignment 3|HW3]], [[14-240/Classnotes for Wednesday October 1|Wednesday]], [[14-240/Tutorial-Sep30|Tutorial]] 
|- align=left
|align=center|5
|Oct 6
|[[14-240/Homework Assignment 4|HW4]], [[14-240/Classnotes for Monday October 6|Monday]], [[14-240/Classnotes for Wednesday October 8|Wednesday]], [[14-240/Tutorial-October7|Tutorial]] 
|- align=left
|align=center|6
|Oct 13
|No Monday class (Thanksgiving), [[14-240/Classnotes for Wednesday October 15|Wednesday]], [[14-240/Tutorial-October14|Tutorial]]
|- align=left
|align=center|7
|Oct 20
|[[14-240/Homework Assignment 5|HW5]], [[14-240/Term Test|Term Test]] at tutorials on Tuesday, [[14-240/Classnotes for Wednesday October 22|Wednesday]]
|- align=left
|align=center|8
|Oct 27
|[[14-240/Homework Assignment 6|HW6]], [[14-240/Classnotes for Monday October 27|Monday]], [[14-240/Linear Algebra - Why We Care|Why LinAlg?]], [[14-240/Classnotes for Wednesday October 29|Wednesday]], [[14-240/Tutorial-October28|Tutorial]]
|- align=left
|align=center|9
|Nov 3
|Monday is the last day to drop this class, [[14-240/Homework Assignment 7|HW7]], [[14-240/Classnotes for Monday November 3|Monday]], [[14-240/Classnotes for Wednesday November 5|Wednesday]]
|- align=left
|align=center|10
|Nov 10
|[[14-240/Homework Assignment 8|HW8]], [[14-240/Classnotes for Monday November 3|Monday]]
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT November break
|- align=left
|align=center|12
|Nov 24
|[[14-240/Homework Assignment 9|HW9]]
|- align=left
|align=center|13
|Dec 1
|Wednesday is a "makeup Monday"!
|- align=left
|align=center|F
|Dec 8
|[[14-240/The Final Exam|Our Final Exam]] will take place on Wednesday December 10, 2-5PM, at GB 404 for students with last names between A and Lo, and at GB 412 for the rest.
|- align=left
|colspan=3 align=center|[[14-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-240-ClassPhoto.jpg|310px|Class Photo]] [[14-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:14-240-Splash.png|310px]]
|}

14-240/Classnotes for Wednesday October 22

2014-10-22T15:41:43Z

Yue.Jiang:

{{14-240/Navigation}}

By Yue Jiang--[[User:Yue.Jiang|Yue.Jiang]] ([[User talk:Yue.Jiang|talk]]) 11:29, 22 October 2014 (EDT)

----

<gallery>
File:Oct 22 note1.jpeg|Oct 22 note1

File:Oct 22 note2.jpg|Oct 22 note2

File:Oct 22 note3.jpg|Oct 22 note3

</gallery>

File:Oct 22 note3.jpg

2014-10-22T15:40:34Z

Yue.Jiang:

File:Oct 22 note2.jpg

2014-10-22T15:40:16Z

Yue.Jiang:

14-240/Classnotes for Wednesday October 22

2014-10-22T15:39:40Z

Yue.Jiang:

{{14-240/Navigation}}

By Yue Jiang--[[User:Yue.Jiang|Yue.Jiang]] ([[User talk:Yue.Jiang|talk]]) 11:29, 22 October 2014 (EDT)

----

<gallery>
File:Oct 22 note1.jpeg|Oct 22 note1

File:Oct 22 note2.jpeg|Oct 22 note2

File:Oct 22 note3.jpeg|Oct 22 note3

</gallery>

14-240/Classnotes for Wednesday October 22

2014-10-22T15:37:01Z

Yue.Jiang:

By Yue Jiang--[[User:Yue.Jiang|Yue.Jiang]] ([[User talk:Yue.Jiang|talk]]) 11:29, 22 October 2014 (EDT)

----

[[File:Oct 22 note1/Oct 22 note1.jpeg]]

[[File:Oct 22 note2]]

[[File:Oct 22 note3]]

File:Oct 22 note1.jpeg

2014-10-22T15:35:29Z

Yue.Jiang:

14-240/Classnotes for Wednesday October 22

2014-10-22T15:29:35Z

Yue.Jiang:

By Yue Jiang--[[User:Yue.Jiang|Yue.Jiang]] ([[User talk:Yue.Jiang|talk]]) 11:29, 22 October 2014 (EDT)

----

[[File:Oct 22 note1]]

[[File:Oct 22 note2]]

[[File:Oct 22 note3]]

14-240/Classnotes for Wednesday October 22

2014-10-22T15:29:01Z

Yue.Jiang: Created page with " ----By Yue Jiang File:Oct 22 note1 File:Oct 22 note2 File:Oct 22 note3"

----By Yue Jiang

[[File:Oct 22 note1]]
[[File:Oct 22 note2]]
[[File:Oct 22 note3]]

14-240/Navigation

2014-10-22T15:25:35Z

Yue.Jiang:

<noinclude>Back to [[14-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center style="color: red;"
|colspan=3|'''Welcome to Math 240!''' 
Term test on Tuesday October 21 at HS 610! Read more '''[[14-240/Term Test|here]]''', also on office hours.
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-240/About This Class|About This Class]], What is this class about? ({{Pensieve link|Classes/14-240/one/What_is_This_Class_AboutQ.pdf|PDF}}, {{Pensieve link|Classes/14-240/What_is_This_Class_AboutQ.html|HTML}}), [[14-240/Classnotes for Monday September 8|Monday]], [[14-240/Classnotes for Wednesday September 10|Wednesday]]
|- align=left
|align=center|2
|Sep 15
|[[14-240/Homework Assignment 1|HW1]], [[14-240/Classnotes for Monday September 15|Monday]], [[14-240/Classnotes for Wednesday September 17|Wednesday]], {{Pensieve link|Classes/14-240/nb/TheComplexField.pdf|TheComplexField.pdf}} 
|- align=left
|align=center|3
|Sep 22
|[[14-240/Homework Assignment 2|HW2]], [[14-240/Class Photo|Class Photo]], [[14-240/Classnotes for Monday September 22|Monday]], [[14-240/Classnotes for Wednesday September 24|Wednesday]] 
|- align=left
|align=center|4
|Sep 29
|[[14-240/Homework Assignment 3|HW3]], [[14-240/Classnotes for Wednesday October 1|Wednesday]], [[14-240/Tutorial-Sep30|Tutorial]] 
|- align=left
|align=center|5
|Oct 6
|[[14-240/Homework Assignment 4|HW4]], [[14-240/Classnotes for Monday October 6|Monday]], [[14-240/Classnotes for Wednesday October 8|Wednesday]], [[14-240/Tutorial-October7|Tutorial]] 
|- align=left
|align=center|6
|Oct 13
|No Monday class (Thanksgiving), [[14-240/Classnotes for Wednesday October 15|Wednesday]], [[14-240/Tutorial-October14|Tutorial]]

|- align=left
|align=center|7
|Oct 20
|[[14-240/Homework Assignment 5|HW5]], [[14-240/Term Test|Term Test]] at tutorials on Tuesday, [[14-240/Classnotes for Wednesday October 22|Wednesday]]
|- align=left
|align=center|8
|Oct 27
|[[14-240/Homework Assignment 6|HW6]]
|- align=left
|align=center|9
|Nov 3
|Monday is the last day to drop this class, [[14-240/Homework Assignment 7|HW7]]
|- align=left
|align=center|10
|Nov 10
|[[14-240/Homework Assignment 8|HW8]]
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT November break, [[14-240/Homework Assignment 9|HW9]]
|- align=left
|align=center|12
|Nov 24
|[[14-240/Homework Assignment 10|HW10]]
|- align=left
|align=center|13
|Dec 1
|Wednesday is a "makeup Monday"!
|- align=left
|align=center|F
|Dec 8
|[[14-240/The Final Exam|Our Final Exam]] will take place on Wednesday December 10, 2-5PM, at GB 404 for students with last names between A and Lo, and at GB 412 for the rest.
|- align=left
|colspan=3 align=center|[[14-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-240-ClassPhoto.jpg|310px|Class Photo]] [[14-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:14-240-Splash.png|310px]]
|}

14-240/Classnotes for Wednesday October 8

2014-10-09T18:30:29Z

Yue.Jiang:

{{14-240/Navigation}}

==Scanned Lecture Notes by [[User Yue.Jiang|Yue Jiang]]==

<gallery>
File:October 8 note 1.jpeg|page 1
File:October 8 note 2.jpeg|page 2
File:October 8 note 3.jpeg|page 3
File:4.jpg|page 4
File:5.jpg|page 5
</gallery>

*http://drorbn.net/images/f/fa/MAT240_%28Oct_8%2C_2014%29_1_of_3.pdf (Notes by AM, 1 of 3)
*http://drorbn.net/images/a/a8/MAT240_%28Oct_8%2C_2014%29_2_of_3.pdf (Notes by AM, 2 of 3)
*http://drorbn.net/images/9/98/MAT240_%28Oct_8%2C_2014%29_3_of_3.pdf (Notes by AM, 3 of 3)

14-240/Classnotes for Wednesday October 8

2014-10-09T15:14:48Z

Yue.Jiang: /* Scanned Lecture Notes by Yue */

File:5.jpg

2014-10-09T15:13:59Z

Yue.Jiang:

14-240/Classnotes for Wednesday October 8

2014-10-09T15:13:41Z

Yue.Jiang:

File:4.jpg

2014-10-09T15:13:07Z

Yue.Jiang:

File:October 8 note 3.jpeg

2014-10-09T15:12:48Z

Yue.Jiang:

File:October 8 note 2.jpeg

2014-10-09T15:12:22Z

Yue.Jiang:

14-240/Classnotes for Wednesday October 8

2014-10-09T15:11:28Z

Yue.Jiang:

14-240/Classnotes for Wednesday October 8

2014-10-09T15:10:26Z

Yue.Jiang:

==Scanned Lecture Notes by [[User Yue.Jiang|Yue]]==

<gallery>
File:October 8 note 1.jpeg|page 1
File:|page 2
</gallery>

File:October 8 note 1.jpeg

2014-10-09T15:04:33Z

Yue.Jiang:

14-240/Classnotes for Wednesday October 8

2014-10-09T15:03:34Z

Yue.Jiang:

[[File:October 8 note]]

14-240/Classnotes for Wednesday October 8

2014-10-09T15:02:41Z

Yue.Jiang: Created page with "Media:1.jpg"

[[Media:1.jpg]]

14-240/Navigation

2014-10-09T14:53:41Z

Yue.Jiang:

<noinclude>Back to [[14-240]]. </noinclude>
{| border="1px" cellpadding="1" cellspacing="0" width="100%" style="font-size: small; align: left"
|- align=center style="color: red;"
|colspan=3|'''Welcome to Math 240!''' 
|- align=left
!#
!Week of...
!Notes and Links
|- align=left
|align=center|1
|Sep 8
|[[14-240/About This Class|About This Class]], What is this class about? ({{Pensieve link|Classes/14-240/one/What_is_This_Class_AboutQ.pdf|PDF}}, {{Pensieve link|Classes/14-240/What_is_This_Class_AboutQ.html|HTML}}), [[14-240/Classnotes for Monday September 8|Monday]], [[14-240/Classnotes for Wednesday September 10|Wednesday]]
|- align=left
|align=center|2
|Sep 15
|[[14-240/Homework Assignment 1|HW1]], [[14-240/Classnotes for Monday September 15|Monday]], [[14-240/Classnotes for Wednesday September 17|Wednesday]], {{Pensieve link|Classes/14-240/nb/TheComplexField.pdf|TheComplexField.pdf}}
|- align=left
|align=center|3
|Sep 22
|[[14-240/Homework Assignment 2|HW2]], [[14-240/Class Photo|Class Photo]], [[14-240/Classnotes for Monday September 22|Monday]], [[14-240/Classnotes for Wednesday September 24|Wednesday]]
|- align=left
|align=center|4
|Sep 29
|[[14-240/Homework Assignment 3|HW3]], [[14-240/Classnotes for Wednesday October 1|Wednesday]], [[14-240/Tutorial-Sep30|Tutorial]]
|- align=left
|align=center|5
|Oct 6
|[[14-240/Homework Assignment 4|HW4]], [[14-240/Classnotes for Monday October 6|Monday]], [[14-240/Classnotes for Wednesday October 8|Wednesday]]
|- align=left
|align=center|6
|Oct 13
|No Monday class (Thanksgiving)
|- align=left
|align=center|7
|Oct 20
|[[14-240/Homework Assignment 5|HW5]], [[14-240/Term Test|Term Test]] at tutorials on Tuesday
|- align=left
|align=center|8
|Oct 27
|[[14-240/Homework Assignment 6|HW6]]
|- align=left
|align=center|9
|Nov 3
|Monday is the last day to drop this class, [[14-240/Homework Assignment 7|HW7]]
|- align=left
|align=center|10
|Nov 10
|[[14-240/Homework Assignment 8|HW8]]
|- align=left
|align=center|11
|Nov 17
|Monday-Tuesday is UofT November break, [[14-240/Homework Assignment 9|HW9]]
|- align=left
|align=center|12
|Nov 24
|[[14-240/Homework Assignment 10|HW10]]
|- align=left
|align=center|13
|Dec 1
|Wednesday is a "makeup Monday"!
|- align=left
|align=center|F
|Dec 8
|[[14-240/The Final Exam|The Final Exam]]?
|- align=left
|align=center|F
|Dec 15
|[[14-240/The Final Exam|The Final Exam]]?
|- align=left
|colspan=3 align=center|[[14-240/Register of Good Deeds|Register of Good Deeds]]
|- align=left
|colspan=3 align=center|[[Image:14-240-ClassPhoto.jpg|310px|Class Photo]] [[14-240/Class Photo|Add your name / see who's in!]]
|- align=left
|colspan=3 align=center|[[Image:14-240-Splash.png|310px]]
|}

14-240/Classnotes for Monday September 22

2014-09-25T03:04:38Z

Yue.Jiang:

Polar coordinates:
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta</math>

The Fundamantal Theorem of Algebra:
<math>a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0</math>
where <math>a_i \in C </math>and<math> a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>z^{2} - 1 = 0</math> has a solution.

* Forces can multiple by a "scalar"(number).
No "multiplication" of forces.

Definition of Vector Space:
A "Vector Space" over a field F is a set V with a special element <math>O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>

s.t.
* <math>VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

14-240/Classnotes for Monday September 22

2014-09-25T03:03:53Z

Yue.Jiang:

Polar coordinates:
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta</math>

The Fundamantal Theorem of Algebra:
<math>a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0</math>
where <math>a_i \in C and a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>z^{2} - 1 = 0</math> has a solution.

* Forces can multiple by a "scalar"(number).
No "multiplication" of forces.

Definition of Vector Space:
A "Vector Space" over a field F is a set V with a special element <math>O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>

s.t.
* <math>VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

14-240/Classnotes for Monday September 22

2014-09-25T03:01:46Z

Yue.Jiang:

Polar coordinates:
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta</math>

The Fundamantal Theorem of Algebra:
<math>\a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0</math>
where <math>\a_i \in C and \a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>z^{2} - 1 = 0</math> has a solution.

* Forces can multiple by a "scalar"(number).
No "multiplication" of forces.

Definition of Vector Space:
A "Vector Space" over a field F is a set V with a special element <math>\O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>

s.t.
* <math>\VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>\VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>\VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>\VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>\VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>\VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>\VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>\VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

14-240/Classnotes for Monday September 22

2014-09-25T03:01:04Z

Yue.Jiang:

Polar coordinates:
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>\r_1 \times e^{i\\theta_2} = r_1 \times (cos\theta + sin\theta</math>

The Fundamantal Theorem of Algebra:
<math>\a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0</math>
where <math>\a_i \in C and \a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>z^{2} - 1 = 0</math> has a solution.

* Forces can multiple by a "scalar"(number).
No "multiplication" of forces.

Definition of Vector Space:
A "Vector Space" over a field F is a set V with a special element <math>\O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>

s.t.
* <math>\VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>\VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>\VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>\VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>\VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>\VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>\VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>\VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

14-240/Classnotes for Monday September 22

2014-09-25T02:55:35Z

Yue.Jiang:

Polar coordinates:
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>\r_1 \times e^{i\\theta_2} = \r_1 \times (cos\theta + sin\theta</math>

The Fundamantal Theorem of Algebra:
<math>\a_n \times z^{n} + \a_n-1 \times z^{n-1} + \dots + \a_0</math>
where <math>\a_i \in C and \a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>z^{2} - 1 = 0</math> has a solution.

* Forces can multiple by a "scalar"(number).
No "multiplication" of forces.

Definition of Vector Space:
A "Vector Space" over a field F is a set V with a special element <math>\O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>

s.t.
* <math>\VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>\VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>\VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>\VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>\VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>\VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>\VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>\VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

14-240/Classnotes for Monday September 22

2014-09-25T02:50:45Z

Yue.Jiang: Created page with "Polar coordinates: * <math>r \times \e^i\theta = r \times cos\theta + i \times rsin\theta</math> * <math>\r_1 \times \e^i\\theta_2 = \r_1 \times (cos\theta + sin\theta</math> ..."

Polar coordinates:
* <math>r \times \e^i\theta = r \times cos\theta + i \times rsin\theta</math>
* <math>\r_1 \times \e^i\\theta_2 = \r_1 \times (cos\theta + sin\theta</math>

The Fundamantal Theorem of Algebra:
<math>\a_n \times \z^n + \a_n-1 \times \z^n-1 + \dots + \a_0</math>
where <math>\a_i \in C and \a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>\z^2 - 1 = 0</math> has a solution.

* Forces can multiple by a "scalar"(number).
No "multiplication" of forces.

Definition of Vector Space:
A "Vector Space" over a field F is a set V with a special element <math>\O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>

s.t.
* <math>\VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>\VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>\VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>\VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>\VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>\VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>\VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>\VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

14-240/Navigation

2014-09-25T01:50:34Z

Yue.Jiang:

14-240/Class Photo

2014-09-25T01:44:25Z

Yue.Jiang: /* Who We Are... */

Our class on September 24, 2014:

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===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First Name
!Last Name
!ID wcashore
!e-mail
!Location
!Comments

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}

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14-240/Class Photo

2014-09-25T01:43:14Z

Yue.Jiang: /* Who We Are... */

Our class on September 24, 2014:

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{| align=center border=1 cellspacing=0
|-
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{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}

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14-240/Class Photo

2014-09-25T01:42:46Z

Yue.Jiang: /* Who We Are... */

Our class on September 24, 2014:

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{| align=center border=1 cellspacing=0
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{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}

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14-240/Class Photo

2014-09-25T01:42:10Z

Yue.Jiang: /* Who We Are... */

Our class on September 24, 2014:

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{| align=center border=1 cellspacing=0
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{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}
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14-240/Classnotes for Wednesday September 10

2014-09-23T21:04:11Z

Yue.Jiang:

{{14-240/Navigation}}
Knowledge about Fields:

During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

===The Real Numbers===

====Properties of Real Numbers====

The real numbers are a set <math>\R</math> with two binary operations:

<math>+ : \R \times \R \rightarrow \R</math>

<math>* : \R \times \R \rightarrow \R</math>

such that the following properties hold.

* R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

* R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

* R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the
| "additive unit" and 1 as the "multiplicative unit")

* R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;
| <math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)

* R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)

====Properties That Do Not Follow from R1-R5====

There are properties which are true for <math>\R</math>, but do not follow from R1 to R5. For example ('''note''' that OR in mathematics denotes an "inclusive or"):
<math>\forall a \in \R, \exists x \in \R, a = x^2</math> OR <math>-a = x^2</math> (the existence of square roots)

Consider another set that satisfies all the properties R1 to R5. In <math>\Q</math> (the rational numbers), let us take </math>a = 2</math>. There is no <math>x \in \Q</math> such that <math>x^2 = a = 2</math>, so the statement above is not true for the rational numbers!

-------------------------------------------------------------------------------------------------------------------------------------------------------

===Fields===

====Definition====

A "field" is a set <math>\mathbb{F}</math> along with a pair of binary operations:
<math>+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

<math>* : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.

* F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

* F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

* F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)

* F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;
| <math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)

* F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)
====Examples====

# <math>\R</math> is a field.
# <math>\Q</math> (the rational numbers) is a field.
# <math>\C</math> (the complex numbers) is a field.
# <math>\mathbb{F} = \{0, 1\}</math> with operations defined as follows (known as <math>\mathbb{F}_2</math> or <math>\Z/2</math>) is a field:

{| class="wikitable"
|-
! +
! 0
! 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 0
|}

{| class="wikitable"
|-
! *
! 0
! 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

 

More generally, for every prime number <math>P</math>, <math>\mathbb{F}_p = \{0, 1, 2, 3, \cdots, p - 1\}</math> is a field, with operations defined by
<math>(a, b) \rightarrow a + b \mod P</math>.

An example: <math>\mathbb{F}_7 = \{0, 1, 2, 3, 4, 5, 6\}</math>, the operations are like remainders when divided by 7, or "like remainders mod 7". For example, <math>4 + 6 = 4 + 6 \mod 7</math> and <math>3 * 5 = 3 * 5 \mod 7</math>.

====Basic Properties of Fields====

'''Theorem''':
Let <math>\mathbb{F}</math> be a field, and let <math>a, b, c</math> denote elements of <math>\mathbb{F}</math>. Then:
# <math>a + b = c + b \Rightarrow a = c</math> (cancellation law)
# <math>b \ne 0 ~\&~ a * b = c * b \Rightarrow a = c</math>
Proof of 1:
1. By F4, <math>\exists b' \in \mathbb{F}, b + b' = 0</math>.
We know that <math>a + b = c + b</math>;
Therefore <math>(a + b) + b' = (c + b) + b'</math>.
2. By F2, <math>a + (b + b') = c + (b + b')</math>,
so by the choice of <math>b'</math>, we know that <math>a + 0 = c + 0</math>.
3. Therefore, by F3, <math>a = c</math>.
＾_＾

Proof of 2: more or less the same.

3. If <math>0' \in \mathbb{F}</math> is "like 0", then it is 0:
If <math>0' \in \mathbb{F}</math> satisfies <math>\forall a \in \mathbb{F}, a + 0' = a</math>, then 0' = 0.

4. If <math>1' \in \mathbb{F}</math> is "like 1", then it is 1:
If <math>1' \in \mathbb{F}</math> satisfies that <math>\forall a \in \mathbb{F}, a * 1' = a</math>, then 1' = 1.

Proof of 3 :
1. By F3 , 0' = 0' + 0.
2. By F1 , 0' + 0 = 0 + 0'.
3. By assumption on 0', 0' = 0 + 0' = 0.
＾_＾

5. <math>\forall a, b, b' \in \mathbb{F}, a + b = 0 ~\&~ a + b' = 0 \Rightarrow b = b'</math>:
In any field "<math>-a</math>" makes sense because it is unique -- it has an unambiguous meaning.
<math>(-a):</math> the <math>b</math> such that <math>a + b = 0</math>.

6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:
In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.

Proof of 5 :
1. <math>a + b = 0 = a + b'</math>.
2. By F1, <math>b + a = b'+ a</math>.
3. By cancellation, <math>b = b'</math>.
＾_＾

7. <math>-(-a) = a</math> and when <math>a \ne 0</math>, <math>(a^{-1})^{-1} = a</math>.

Proof of 7 :
1. By definition, <math>a + (-a) = 0</math>. (*)
2. By definition, <math>(-a) + (-(-a) = 0</math>.
3. By (*) and F1, <math>(-a) + a = 0</math>.
4. By property 5, <math>-(-a) = a</math>.
＾_＾

===Scanned Lecture Notes by [[User:AM|AM]]===

<gallery>
File:MAT 240 (1 of 2) Sept 10, 2014.pdf‎|page 1
File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎|page 2
</gallery>

14-240/Classnotes for Monday September 15

2014-09-18T03:43:33Z

Yue.Jiang:

{{14-240/Navigation}}
Definition:
* Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
* Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.

Theorem:

* 8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
*proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>
By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
By F3 , <math>a \times 0 = 0 + a \times 0</math>;
By Thm P1,<math>0 = a \times 0</math>.

* 9. <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
proof of 9: By F3 , <math>\times b = 0 \neq 1</math>.

* 10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.

* 11. <math>(-a) \times (-b) = a \times b</math>.

* 12. <math>a \times b = 0 \iff a = 0 or b = 0</math>.
proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
=> : Assume <math>a \times b = 0 </math> , if a = 0 we are done;
Otherwise , by P8 , <math>a \neq 0 </math> and we have <math>a \times b = 0 = a \times 0</math>;
by cancellation (P2) , <math>b = 0</math>.

<math>(a + b) \times (a - b) = a^2 - b^2</math>.
proof: By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))</math>
<math>= a \times a + a \times (-b) + b \times a + (-b) \times b</math>
<math>= a^2 - b^2</math>
Theorem :
<math>\exists! \iota : \Z \rightarrow F</math> s.t.
1. <math>\iota(0) = 0 , \iota(1) = 1</math>;
2. <math>\forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n)</math>;
3. <math>\forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n)</math>.

<math>\iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1;</math>
<math>\iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1;</math>

......

In F2 , <math>27 ----> \iota(27) = \iota(26 + 1)</math>
<math>= \iota(26) + \iota(1)</math>
<math>= \iota(26) + 1</math>
<math>= \iota(13 \times 2) + 1</math>
<math>= \iota(2) \times \iota(13) + 1</math>
<math>= (1 + 1) \times \iota(13) + 1</math>
<math>= 0 \times \iota(13) + 1</math>
<math>= 1</math>
http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)
http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)

14-240/Classnotes for Monday September 15

2014-09-15T16:03:19Z

Yue.Jiang:

Definition:
Subtract: if <math>a , b </math> belong to <math>F , a - b = a + (-b)</math>.
Divition: if <math>a , b </math> belong to <math>F , a / b = a * (b </math>to the power <math>(-1)</math>.

Theorem:

8. For every <math>a</math> belongs to F , <math>a * 0 = 0</math>.
proof of 8: By F3 , <math>a * 0 = a * (0 + 0)</math>;
By F5 , <math>a * (0 + 0) = a * 0 + a * 0</math>;
By F3 , <math>a * 0 = 0 + a * 0</math>;
By Thm P1 ,<math>0 = a * 0</math>.

9. There not exists <math>b</math> belongs to F s.t. <math>0 * b = 1</math>;
For every <math>b</math> belongs to F s.t. <math>0 * b </math>is not equal to <math>1</math>.
proof of 9: By F3 , <math>0 * b = 0 </math>is not equal to <math>1</math>.

10. <math>(-a) * b = a * (-b) = -(a * b)</math>.

11. <math>(-a) * (-b) = a * b</math>.

12. <math>a * b = 0 iff a = 0 or b = 0</math>.
proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a * b = 0 * b = 0</math>;
By P8 , if <math>b = 0</math> , then <math>a * b = a * 0 = 0</math>.
=> : Assume <math>a * b = 0 </math> , if a = 0 we have done;
Otherwise , by P8 , <math>a </math> is not equal to <math>0 </math>and we have <math>a * b = 0 = a * 0</math>;
by cancellation (P2) , <math>b = 0</math>.

<math>(a + b) * (a - b) = a square - b square</math>.
proof: By F5 , <math>(a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b))
= a * a + a * (-b) + b * a + (-b) * b
= a square - b square</math>
Theorem :
There exists !(unique) <math>iota : Z ---> F</math> s.t.
1. <math>iota(0) = 0 , iota(1) = 1</math>;
2. For every <math>m ,n</math> belong to <math>Z</math> , <math>iota(m+n) = iota(m) + iota(n)</math>;
3. >For every <math>m ,n</math> belong to <math>Z</math> , <math>iota(m*n) = iota(m) * iota(n)</math>.

iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1;
iota(3) = iota(2+1) = iota(2) + iota(1) = iota(2) + 1;
......

In F2 , <math>27 ----> iota(27) = iota(26 + 1)
= iota(26) + iota(1)
= iota(26) + 1
= iota(13 * 2) + 1
= iota(2) * iota(13) + 1
= (1 + 1) * iota(13) + 1
= 0 * iota(13) + 1
= 1</math>

14-240/Classnotes for Monday September 15

2014-09-15T16:01:45Z

Yue.Jiang:

Definition:
Subtract: if <math>a , b </math> belong to <math>F , a - b = a + (-b)</math>.
Divition: if <math>a , b </math> belong to F , <math>a / b = a * (b </math>to the power <math>(-1)</math>.

Theorem:

8. For every <math>a</math> belongs to F , <math>a * 0 = 0</math>.
proof of 8: By F3 , <math>a * 0 = a * (0 + 0)</math>;
By F5 , <math>a * (0 + 0) = a * 0 + a * 0</math>;
By F3 , <math>a * 0 = 0 + a * 0</math>;
By Thm P1 ,<math>0 = a * 0</math>.

9. There not exists <math>b</math> belongs to F s.t. <math>0 * b = 1</math>;
For every <math>b</math> belongs to F s.t. <math>0 * b </math>is not equal to <math>1</math>.
proof of 9: By F3 , <math>0 * b = 0 </math>is not equal to <math>1</math>.

10. <math>(-a) * b = a * (-b) = -(a * b)</math>.

11. <math>(-a) * (-b) = a * b</math>.

12. <math>a * b = 0 iff a = 0 or b = 0</math>.
proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a * b = 0 * b = 0</math>;
By P8 , if <math>b = 0</math> , then <math>a * b = a * 0 = 0</math>.
=> : Assume <math>a * b = 0 </math> , if a = 0 we have done;
Otherwise , by P8 , <math>a </math> is not equal to <math>0 </math>and we have <math>a * b = 0 = a * 0</math>;
by cancellation (P2) , <math>b = 0</math>.

<math>(a + b) * (a - b) = a square - b square</math>.
proof: By F5 , <math>(a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b))
= a * a + a * (-b) + b * a + (-b) * b
= a square - b square</math>
Theorem :
There exists !(unique) <math>iota : Z ---> F</math> s.t.
1. <math>iota(0) = 0 , iota(1) = 1</math>;
2. For every <math>m ,n</math> belong to Z , <math>iota(m+n) = iota(m) + iota(n)</math>;
3. >For every <math>m ,n</math> belong to Z , <math>iota(m*n) = iota(m) * iota(n)</math>.

iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1;
iota(3) = iota(2+1) = iota(2) + iota(1) = iota(2) + 1;
......

In F2 , <math>27 ----> iota(27) = iota(26 + 1)
= iota(26) + iota(1)
= iota(26) + 1
= iota(13 * 2) + 1
= iota(2) * iota(13) + 1
= (1 + 1) * iota(13) + 1
= 0 * iota(13) + 1
= 1</math>

14-240/Classnotes for Monday September 15

2014-09-15T16:00:46Z

Yue.Jiang:

Definition:
Subtract: if <math>a , b </math> belong to <math>F , a - b = a + (-b)</math>.
Divition: if <math>a , b </math> belong to F , <math>a / b = a * (b to the power (-1)</math>.

Theorem:

8. For every <math>a belongs to F , a * 0 = 0</math>.
proof of 8: By F3 , <math>a * 0 = a * (0 + 0)</math>;
By F5 , <math>a * (0 + 0) = a * 0 + a * 0</math>;
By F3 , <math>a * 0 = 0 + a * 0</math>;
By Thm P1 ,<math>0 = a * 0</math>.

9. There not exists <math>b</math> belongs to F s.t. <math>0 * b = 1</math>;
For every <math>b</math> belongs to F s.t. <math>0 * b </math>is not equal to <math>1</math>.
proof of 9: By F3 , <math>0 * b = 0 </math>is not equal to <math>1</math>.

10. <math>(-a) * b = a * (-b) = -(a * b)</math>.

11. <math>(-a) * (-b) = a * b</math>.

12. <math>a * b = 0 iff a = 0 or b = 0</math>.
proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a * b = 0 * b = 0</math>;
By P8 , if <math>b = 0</math> , then <math>a * b = a * 0 = 0</math>.
=> : Assume <math>a * b = 0 </math> , if a = 0 we have done;
Otherwise , by P8 , <math>a </math> is not equal to <math>0 </math>and we have <math>a * b = 0 = a * 0</math>;
by cancellation (P2) , <math>b = 0</math>.

<math>(a + b) * (a - b) = a square - b square</math>.
proof: By F5 , <math>(a + b) * (a - b) = a * (a + (-b)) + b * (a + (-b))
= a * a + a * (-b) + b * a + (-b) * b
= a square - b square</math>
Theorem :
There exists !(unique) <math>iota : Z ---> F</math> s.t.
1. <math>iota(0) = 0 , iota(1) = 1</math>;
2. For every <math>m ,n</math> belong to Z , <math>iota(m+n) = iota(m) + iota(n)</math>;
3. >For every <math>m ,n</math> belong to Z , <math>iota(m*n) = iota(m) * iota(n)</math>.

iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1;
iota(3) = iota(2+1) = iota(2) + iota(1) = iota(2) + 1;
......

In F2 , <math>27 ----> iota(27) = iota(26 + 1)
= iota(26) + iota(1)
= iota(26) + 1
= iota(13 * 2) + 1
= iota(2) * iota(13) + 1
= (1 + 1) * iota(13) + 1
= 0 * iota(13) + 1
= 1</math>