14-240/Classnotes for Monday September 15

DBN: Classes: 2014-15: 14-240 / Navigation

#	Week of...	Notes and Links
Welcome to Math 240! (additions to this web site no longer count towards good deed points)
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
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Definition:

Subtraction: if $a,b\in F,a-b=a+(-b)$ .
Division: if $a,b\in F,a/b=a\times b^{-1}$ .

Theorem:

8. $\forall a\in F$ , $a\times 0=0$ .

                   proof of 8: By F3 ,  $a\times 0=a\times (0+0)$ ;
                               By F5 ,  $a\times (0+0)=a\times 0+a\times 0$ ;
                               By F3 ,  $a\times 0=0+a\times 0$ ;
                               By Thm P1, $0=a\times 0$ .
       
        9.  $\nexists b\in F$  s.t.  $0\times b=1$ ;
            $\forall b\in F$  s.t.  $0\times b\neq 1$ .
                   proof of 9: By F3 ,  $\times b=0\neq 1$ .
       
       10.  $(-a)\times b=a\times (-b)=-(a\times b)$ .
     
       11.  $(-a)\times (-b)=a\times b$ .
      
       12.  $a\times b=0\iff a=0orb=0$ .
                   proof of 12: <= : By P8 , if  $a=0$  , then  $a\times b=0\times b=0$ ;
                                     By P8 , if  $b=0$  , then  $a\times b=a\times 0=0$ .
                                => : Assume  $a\times b=0$  , if a = 0 we are done;
                                     Otherwise , by P8 ,  $a\neq 0$  and we have  $a\times b=0=a\times 0$ ;  
                                                 by cancellation (P2) ,  $b=0$ .

$(a+b)\times (a-b)=a^{2}-b^{2}$ .

        proof: By F5 ,  $(a+b)\times (a-b)=a\times (a+(-b))+b\times (a+(-b))$ 
                                                $=a\times a+a\times (-b)+b\times a+(-b)\times b$ 
                                                $=a^{2}-b^{2}$

Theorem :

         $\exists !\iota :\mathbb {Z} \rightarrow F$   s.t.
              1.  $\iota (0)=0,\iota (1)=1$ ;
              2.  $\forall m,n\in \mathbb {Z} ,\iota (m+n)=\iota (m)+\iota (n)$ ;
              3.  $\forall m,n\in \mathbb {Z} ,\iota (m\times n)=\iota (m)\times \iota (n)$ .

         $\iota (2)=\iota (1+1)=\iota (1)+\iota (1)=1+1;$ 
         $\iota (3)=\iota (2+1)=\iota (2)+\iota (1)=\iota (2)+1;$

        ......                                                                          
     
        In F2 ,  $27---->\iota (27)=\iota (26+1)$ 
                                         $=\iota (26)+\iota (1)$ 
                                         $=\iota (26)+1$ 
                                         $=\iota (13\times 2)+1$ 
                                         $=\iota (2)\times \iota (13)+1$ 
                                         $=(1+1)\times \iota (13)+1$ 
                                         $=0\times \iota (13)+1$ 
                                         $=1$