14-240/Classnotes for Monday September 15

DBN: Classes: 2014-15: 14-240 / Navigation Welcome to Math 240! (additions to this web site no longer count towards good deed points) # Week of... Notes and Links 1 Sep 8 About This Class, What is this class about? (PDF, HTML), Monday, Wednesday 2 Sep 15 HW1, Monday, Wednesday, TheComplexField.pdf,HW1_solutions.pdf 3 Sep 22 HW2, Class Photo, Monday, Wednesday, HW2_solutions.pdf 4 Sep 29 HW3, Wednesday, Tutorial, HW3_solutions.pdf 5 Oct 6 HW4, Monday, Wednesday, Tutorial, HW4_solutions.pdf 6 Oct 13 No Monday class (Thanksgiving), Wednesday, Tutorial 7 Oct 20 HW5, Term Test at tutorials on Tuesday, Wednesday 8 Oct 27 HW6, Monday, Why LinAlg?, Wednesday, Tutorial 9 Nov 3 Monday is the last day to drop this class, HW7, Monday, Wednesday, Tutorial 10 Nov 10 HW8, Monday, Tutorial 11 Nov 17 Monday-Tuesday is UofT November break 12 Nov 24 HW9 13 Dec 1 Wednesday is a "makeup Monday"! End-of-Course Schedule, Tutorial F Dec 8 The Final Exam Register of Good Deeds Add your name / see who's in!

Definition of Subtraction and Division

• Subtraction: if [math]\displaystyle{ a, b \in F, a - b = a + (-b) }[/math].
• Division: if [math]\displaystyle{ a, b \in F, a / b = a \times b^{-1} }[/math].

Basic Properties of a Field (cont'd)

8. [math]\displaystyle{ \forall a \in F }[/math], [math]\displaystyle{ a \times 0 = 0 }[/math].

Proof of 8

By F3 , [math]\displaystyle{ a \times 0 = a \times (0 + 0) }[/math]

By F5 , [math]\displaystyle{ a \times (0 + 0) = a \times 0 + a \times 0 }[/math];

By F3 , [math]\displaystyle{ a \times 0 = 0 + a \times 0 }[/math];

By Thm P1, [math]\displaystyle{ 0 = a \times 0 }[/math].

9. [math]\displaystyle{ \nexists b \in F }[/math] s.t. [math]\displaystyle{ 0 \times b = 1 }[/math];

[math]\displaystyle{ \forall b \in F }[/math] s.t. [math]\displaystyle{ 0 \times b \neq 1 }[/math].

Proof of 9

By F3 , [math]\displaystyle{ \times b = 0 \neq 1 }[/math].

10. [math]\displaystyle{ (-a) \times b = a \times (-b) = -(a \times b) }[/math].

11. [math]\displaystyle{ (-a) \times (-b) = a \times b }[/math].

12. [math]\displaystyle{ a \times b = 0 \iff a = 0 }[/math] or [math]\displaystyle{ b = 0 }[/math].

Proof of 12

<= :

By P8 , if [math]\displaystyle{ a = 0 }[/math] , then [math]\displaystyle{ a \times b = 0 \times b = 0 }[/math];

By P8 , if [math]\displaystyle{ b = 0 }[/math] , then [math]\displaystyle{ a \times b = a \times 0 = 0 }[/math].

=> : Assume [math]\displaystyle{ a \times b = 0 }[/math] , if a = 0 we are done;

Otherwise , by P8 , [math]\displaystyle{ a \neq 0 }[/math] and we have [math]\displaystyle{ a \times b = 0 = a \times 0 }[/math];

by cancellation (P2) , [math]\displaystyle{ b = 0 }[/math].

[math]\displaystyle{ (a + b) \times (a - b) = a^2 - b^2 }[/math].

Proof

By F5 , [math]\displaystyle{ (a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b)) }[/math]

[math]\displaystyle{ = a^2 - b^2 }[/math]

Theorem

[math]\displaystyle{ \exists! \iota : \Z \rightarrow F }[/math] s.t.

1. [math]\displaystyle{ \iota(0) = 0 , \iota(1) = 1 }[/math];

2. [math]\displaystyle{ \forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n) }[/math];

3. [math]\displaystyle{ \forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n) }[/math].

Examples

[math]\displaystyle{ \iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1; }[/math] [math]\displaystyle{ \iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1; }[/math]

......

In F2: [math]\displaystyle{ \begin{align} 27 ----\gt \iota(27) &= \iota(26 + 1)\\ &= \iota(26) + \iota(1)\\ &= \iota(26) + 1\\ &= \iota(13 \times 2) + 1\\ &= \iota(2) \times \iota(13) + 1\\ &= (1 + 1) \times \iota(13) + 1\\ &= 0 \times \iota(13) + 1\\ &= 1 \end{align} }[/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)