Drorbn - User contributions [en]

User:R.mcclure

2014-10-01T20:00:49Z

R.mcclure: Created page with "Apparently I can do some sort of maths. yay?"

Apparently I can do some sort of maths.

yay?

14-240/Class Photo

2014-10-01T20:00:09Z

R.mcclure: /* Who We Are... */

Our class on September 24, 2014:

[[Image:14-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{14-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

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

{| align=center border=1 cellspacing=0
|-
!First Name
!Last Name
!ID
!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.}}



{{Photo Entry|last=Aleem|first=Asad|userid=AsadAleem|email=asad.aleem@ mail.utoronto.ca|location=Third from left in the seventh or eighth row, wearing a bright blue T-Shirt|comments=}}

{{Photo Entry|last=An|first=Ruiwen|userid=Christine An|email=christine.an@ mail.utoronto.ca|location=The "sunny girl" in dark brown (maybe) fifth from the left in the second row|comments=}}

{{Photo Entry|last=Field|first=Grace|userid=Grace.field|email=grace.field@ mail.utoronto.ca|location=The girl wearing a dark blue shirt in the third row, fifth from right|comments=}}

{{Photo Entry|last=Gomes|first=Andrew|userid=Agomes|email=andrew.gomes@ mail. utoronto. ca|location=The "young man" in the second row wearing a white T-shirt|comments=}}

{{Photo Entry|last=Huang|first=Charles|userid=Charlesh|email=cherls.huang@ mail. utoronto. ca|location=Fifth row, far left, stripped shirt|comments=}}

{{Photo Entry|last=Jiang|first=Yue|userid=Yue.Jiang|email=yuenj.jiang@ mail. utoronto. ca|location=The "little girl" in the second row, second from the left|comments=}}

{{Photo Entry|last=Luo|first=Danny (Xiao)|userid=Danny.luo|email=danny.luo@ mail.utoronto.ca|location=The man third from the left in the second row, with a backpack in between his legs|comments=}}

{{Photo Entry|last=McClure|first=Robertson|userid=r.mcclure|email=r.mcclure@ mail. toronto. edu|location= Sitting in the top left in a baby blue t-shirt wearing lanyard and small black necklace|comments=}}

{{Photo Entry|last=Shen|first=Crystal|userid=tsodssy|email=crystal.shen@ mail. utoronto. ca|location=The girl not facing the camera (How did this happen?) in the front row right corner|comments=}}

{{Photo Entry|last=Shim|first=Soho|userid=Soho|email=soho.shim@ mail. utoronto. ca|location=The girl wearing a white T-shirt in the first row, third from the right|comments=}}

{{Photo Entry|last=Tjandra|first=Donna|userid=Donna Tjandra|email=donna.tjandra@ mail.utoronto.ca|location=The girl in the 9th full row, 4th from the right, wearing a purple sweater|comments=}}

{{Photo Entry|last=Wei|first=Zhiyang|userid=Gianne|email=zhiyang.wei@ mail.utoronto.ca|location=second row, third from the right|comments=}}

{{Photo Entry|last=Wu|first=Boyang|userid=Boyang.wu|email=boyang.wu@mail.utoronto.ca|location=The boy with plaid shirt in the middle of fourth row from bottom and no neighbor sits beside him.|comments=}}

|}

14-240/Class Photo

2014-10-01T19:59:09Z

R.mcclure: /* Who We Are... */

Our class on September 24, 2014:

[[Image:14-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{14-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

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

{| align=center border=1 cellspacing=0
|-
!First Name
!Last Name
!ID
!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.}}



{{Photo Entry|last=Aleem|first=Asad|userid=AsadAleem|email=asad.aleem@ mail.utoronto.ca|location=Third from left in the seventh or eighth row, wearing a bright blue T-Shirt|comments=}}

{{Photo Entry|last=An|first=Ruiwen|userid=Christine An|email=christine.an@ mail.utoronto.ca|location=The "sunny girl" in dark brown (maybe) fifth from the left in the second row|comments=}}

{{Photo Entry|last=Field|first=Grace|userid=Grace.field|email=grace.field@ mail.utoronto.ca|location=The girl wearing a dark blue shirt in the third row, fifth from right|comments=}}

{{Photo Entry|last=Gomes|first=Andrew|userid=Agomes|email=andrew.gomes@ mail. utoronto. ca|location=The "young man" in the second row wearing a white T-shirt|comments=}}

{{Photo Entry|last=Huang|first=Charles|userid=Charlesh|email=cherls.huang@ mail. utoronto. ca|location=Fifth row, far left, stripped shirt|comments=}}

{{Photo Entry|last=Jiang|first=Yue|userid=Yue.Jiang|email=yuenj.jiang@ mail. utoronto. ca|location=The "little girl" in the second row, second from the left|comments=}}

{{Photo Entry|last=Luo|first=Danny (Xiao)|userid=Danny.luo|email=danny.luo@ mail.utoronto.ca|location=The man third from the left in the second row, with a backpack in between his legs|comments=}}

{{Photo Entry|last=McClure|first=Robertson|userid=McClureR|email=r.mcclure@ mail. toronto. edu|location= Sitting in the top left in a baby blue t-shirt|comments=}}

{{Photo Entry|last=Shen|first=Crystal|userid=tsodssy|email=crystal.shen@ mail. utoronto. ca|location=The girl not facing the camera (How did this happen?) in the front row right corner|comments=}}

{{Photo Entry|last=Shim|first=Soho|userid=Soho|email=soho.shim@ mail. utoronto. ca|location=The girl wearing a white T-shirt in the first row, third from the right|comments=}}

{{Photo Entry|last=Tjandra|first=Donna|userid=Donna Tjandra|email=donna.tjandra@ mail.utoronto.ca|location=The girl in the 9th full row, 4th from the right, wearing a purple sweater|comments=}}

{{Photo Entry|last=Wei|first=Zhiyang|userid=Gianne|email=zhiyang.wei@ mail.utoronto.ca|location=second row, third from the right|comments=}}

{{Photo Entry|last=Wu|first=Boyang|userid=Boyang.wu|email=boyang.wu@mail.utoronto.ca|location=The boy with plaid shirt in the middle of fourth row from bottom and no neighbor sits beside him.|comments=}}

|}

14-240/Classnotes for Monday September 8

2014-09-11T19:32:05Z

R.mcclure:

{{14-240/Navigation}}
We went over "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}}), then over "[[14-240/About This Class|About This Class]]", and then over the first few properties of real numbers that we will care about.

{{14-240:Dror/Students Divider}}

The real numbers a set '''R'''
with 2 binary operations +, *

+:'''R'''*'''R'''→'''R
'''
*:'''R'''*'''R'''→'''R
'''
in addition 2 special element 0,1∈ '''R'''
s.t. 0≠1 & furthermore:

R1: The commutative law (for both addition and multiplication)

For every a,b∈'''R''', we have
a+b=b+a & ab=ba

R2: The associative law

For every a,b,c∈'''R''', we have
(a+b)+c=a+(b+c)
(ab)c=a(bc)

in our lives
pretty little girls
(PL)G≠P(LG)

R3: a+0=a & a*1=a

== Wednesday September 10th 2014 - Fields ==

The real numbers: A set |R with +,x : |R x |R -> |R & <math>0=/=1</math> are elements of |R
such that

R1: For every <math>a, b</math> that are elements of |R , <math>a + b = b + a</math> and <math>ab = ba </math>

R2: For every <math>a, b, c</math> that are elements of |R, <math>( a + b ) + c = a + ( b + c )</math> and <math> (ab)c = a(bc)</math>

R3: For every <math>a</math> that is an element of |R, <math>a + 0 = a</math> and <math> a * 1 = a</math>

R4: For every a that is an element of |R there exists b that is an element of |R such that <math>a + b = 0</math>
& for every a that is an element of |R and <math>a =/= 0</math> there exists b that is an element |R such that <math>a * b = 1</math>

R5: For every <math>a, b, c</math> that are elements of |R, <math>( a + b ) c = ac + bc</math>

<math>( a + b ) * ( a - b ) = a^2 - b^2</math> follows from R1-R5

The following is true for the Real Numbers but does not follow from R1-R5
For every a that is an element of |R there exists an <math>x</math> that is an element of |R such that <math>a = x^2 or a + x^2 = 0</math>

However we can see that it does not follow from R1-R5 because we can find a field that obeys R1-R5 yet does not follow the above rule.
An example of this is the Rational Numbers |Q. In |Q take <math>a = 2</math> and there does not exist <math>x</math> such that <math>2 = x^2</math> or <math>2 + x^2 = 0</math>

The Definition Of A Field:
A "Field" is a set F along with a pair of binary operations +,x : FxF -> F and along with a pair <math>0, 1</math> that are elements of F such that <math>0 =/= 1</math> and such that R1-R5 hold.

R1: For every <math>a, b</math> that are elements of F , <math>a + b = b + a</math> and <math> ab = ba</math>

R2: For every <math>a, b, c</math> that are elements of F, <math>( a + b ) + c = a + ( b + c )</math> and <math>(ab)c = a(bc)</math>

R3: For every <math>a</math> that is an element of F, <math>a + 0 = a</math> and <math>a * 1 = a</math>

R4: For every <math>a</math> that is an element of F there exists <math>b</math> that is an element of F such that <math>a + b = 0</math>
& for every <math>a</math> that is an element of F and <math>a =/= 0</math> there exists <math>b</math> that is an element F such that <math>a * b = 1</math>

R5: For every <math>a, b, c</math> that are elements of F, <math>( a + b ) c = ac + bc</math>

Example

1. |R is a field (real numbers)
2. |Q is a field (rational numbers)
3. |C is a field (complex numbers)
4. F = {0, 1}

*insert table of addition and multiplication*

Proposition: F is a Field
checking F5

etc...

F = {0 , 1} = F2 = Z/2

Do the same for F7

*insert table of addition and multiplication*

"Like remainders when you divide by 7"
"like remainders mod 7'

Theorem (that shall remain unproved) :
For every prime number P, FP = {0 , 1 , 2 , ... , p-1 }
along with + & x defined as above
<math>( a , b ) -> a + b mod p</math>
is a field.

----

Theorem: (basic properties of Fields)

Let F be a Field, and let a , b , c denote elements of F
Then:
1. <math>a + b = c + b -> a = c </math>
"Cancellation" still holds
2. <math>b =/= 0 , ab = cb -> a = c</math>
3. If <math>0'</math> is an element of F and satisfies for every <math>a , a + 0' = a</math> , then <math>0' = 0 </math>
4. If <math>1'</math> is "like 1" then <math>1' = 1</math>

... to be continued...

14-240/Classnotes for Monday September 8

2014-09-11T16:19:28Z

R.mcclure: The first half of Wednesday's Lesson

{{14-240/Navigation}}
We went over "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}}), then over "[[14-240/About This Class|About This Class]]", and then over the first few properties of real numbers that we will care about.

{{14-240:Dror/Students Divider}}

The real numbers a set '''R'''
with 2 binary operations +, *

+:'''R'''*'''R'''→'''R
'''
*:'''R'''*'''R'''→'''R
'''
in addition 2 special element 0,1∈ '''R'''
s.t. 0≠1 & furthermore:

R1: The commutative law (for both addition and multiplication)

For every a,b∈'''R''', we have
a+b=b+a & ab=ba

R2: The associative law

For every a,b,c∈'''R''', we have
(a+b)+c=a+(b+c)
(ab)c=a(bc)

in our lives
pretty little girls
(PL)G≠P(LG)

R3: a+0=a & a*1=a

Wednesday September 10th 2014 - Fields

The real numbers: A set |R with +,x : |R x |R -> |R & 0=/=1 are elements of |R
such that

R1: For every a, b that are elements of |R , a + b = b + a
& ab = ba

R2: For every a, b, c that are elements of |R, ( a + b ) + c = a + ( b + c )
& (ab)c = a(bc)

R3: For every a that is an element of |R, a + 0 = a
& a * 1 = a

R4: For every a that is an element of |R there exists b that is an element of |R such that a + b = 0
& for every a that is an element of |R and a =/= 0 there exists b that is an element |R such that a * b = 1

R5: For every a, b, c that are elements of |R, ( a + b ) c = ac + bc

( a + b ) * ( a - b ) = a^2 - b^2 follows from R1-R5

The following is true for the Real Numbers but does not follow from R1-R5
For every a that is an element of |R there exists an x that is an element of |R such that a = x^2 or a + x^2 = 0

However we can see that it does not follow from R1-R5 because we can find a field that obeys R1-R5 yet does not follow the above rule.
An example of this is the Rational Numbers |Q. In |Q take a = 2 and there does not exist x such that 2 = x^2 or 2 + x^2 = 0

The Definition Of A Field:
A "Field" is a set F along with a pair of binary operations +,x : FxF -> F and along with a pair 0, 1 that are elements of F such that 0 =/= 1 & such that R1-R5 hold.

R1: For every a, b that are elements of F , a + b = b + a
& ab = ba

R2: For every a, b, c that are elements of F, ( a + b ) + c = a + ( b + c )
& (ab)c = a(bc)

R3: For every a that is an element of F, a + 0 = a
& a * 1 = a

R4: For every a that is an element of F there exists b that is an element of F such that a + b = 0
& for every a that is an element of F and a =/= 0 there exists b that is an element F such that a * b = 1

R5: For every a, b, c that are elements of F, ( a + b ) c = ac + bc

Example

1. |R is a field (real numbers)
2. |Q is a field (rational numbers)
3. |C is a field (complex numbers)
4. F = {0, 1}

*insert table of addition and multiplication*

Proposition: F is a Field
checking F5

a , b , c | ( a + b ) c | ab + bc | good?
0 , 0 , 0 | 0 | 0 | yes

etc...

F = {0 , 1} = F2 = Z/2

Do the same for F7

*insert table of addition and multiplication*

"Like remainders when you divide by 7"
"like remainders mod 7'

Theorem (that shall remain unproved) :
For every prime number P, FP = {0 , 1 , 2 , ... , p-1 }
along with + & x defined as above
( a , b ) -> a + b mod p
is a field.

----

Theorem: (basic properties of Fields)

Let F be a Field, and let a , b , c denote elements of F
Then:
1. a + b = c + b -> a = c
"Cancellation" still holds
2. b =/= 0 , ab = cb -> a = c
3. If 0' is an element of F and satisfies for every a , a + 0' = a , then 0' = 0
4. If 1' is "like 1" then 1' = 1

... to be continued...