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

Basic Properties of a Field (cont'd)

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. $a\times b=0\iff a=0$ or $b=0$.

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 , $(a+b)\times (a-b)=a\times (a+(-b))+b\times (a+(-b))$

$=a^{2}-b^{2}$

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. $\forall m,n\in \mathbb {Z} ,\iota (m\times n)=\iota (m)\times \iota (n)$.

Examples

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

......

In F2: ${\begin{aligned}27---->\iota (27)&=\iota (26+1)\\&=\iota (26)+\iota (1)\\&=\iota (26)+1\\&=\iota (13\times 2)+1\\&=\iota (2)\times \iota (13)+1\\&=(1+1)\times \iota (13)+1\\&=0\times \iota (13)+1\\&=1\end{aligned}}$

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