09-240/Classnotes for Thursday September 17

09-240/Navigation Panel   Additions to the MAT 240 web site no longer count towards good deed points # Week of... Notes and Links 1 Sep 7 Tue, About, Thu 2 Sep 14 Tue, HW1, HW1 Solution, Thu 3 Sep 21 Tue, HW2, HW2 Solution, Thu, Photo 4 Sep 28 Tue, HW3, HW3 Solution, Thu 5 Oct 5 Tue, HW4, HW4 Solution, Thu, 6 Oct 12 Tue, Thu 7 Oct 19 Tue, HW5, HW5 Solution, Term Test on Thu 8 Oct 26 Tue, Why LinAlg?, HW6, HW6 Solution, Thu 9 Nov 2 Tue, MIT LinAlg, Thu 10 Nov 9 Tue, HW7, HW7 Solution Thu 11 Nov 16 Tue, HW8, HW8 Solution, Thu 12 Nov 23 Tue, HW9, HW9 Solution, Thu 13 Nov 30 Tue, On the final, Thu S Dec 7 Office Hours F Dec 14 Final on Dec 16 To Do List The Algebra Song! Register of Good Deeds Misplaced Material Add your name / see who's in!

NSERC - CMS Math in Moscow Scholarships

The Natural Sciences and Engineering Research Council (NSERC) and the Canadian Mathematical Society (CMS) support scholarships at $9,000 each. Canadian students registered in a mathematics or computer science program are eligible.

The scholarships are to attend a semester at the small elite Moscow Independent University.

Math in Moscow Program http://www.mccme.ru/mathinmoscow/

Application details http://www.cms.math.ca/Scholarships/Moscow

For additional information please see your department or call the CMS at 613-733-2662.

Deadline September 30, 2009 to attend the Winter 2010 semester.

Some links

• Dori Eldar's work on "mechanical computations": Machines as Calculating Devices and Computing the function $W=Z^2$ the hard way.
• The "Dimensions" video on "Nombres complexes", is at http://dimensions-math.org/Dim_reg_AM.htm (and then go to "Dimensions_5".

Class notes for today

• Convention for today: $x,y,a,b,c,d,...$ will be real numbers; $z,w,u,v,...$ will be complex numbers

Dream: Find a field $\mathbb C$ that contains $\mathbb R$ and also contains an element $i$ such that $i^2=-1$

Implications:

• $b \in \mathbb R \Rightarrow bi \in \mathbb C$

• $a \in \mathbb R \Rightarrow a+bi \in \mathbb C$

• $c,d \in \mathbb R \Rightarrow c+di \in \mathbb C$

• $\Rightarrow (a+bi)+(c+di)$ must be in $\mathbb C$

$=(a+c)+(bi+di)$

$=(a+c)+(b+d)i$

$=e+fi$

$(a+bi)(c+di)=(a+c)+(b+d)i$

$=a(c+di)+bi(c+di)$

$=ac+adi+bic+bidi$

$=ac+bdi^2 + adi+bci$

$=(ac-bd)+(ad+bc)i$

$=e+fi$

$0_C=0+0i$

$1_C=1+0i$

$(a+bi)+(c+di)=0+0i$

$-(a+bi)=(-a)+(-b)i$

$a+bi \neq 0 \Rightarrow (a,b) \neq 0$

• Find another element of $\mathbb C$, $x+yi$ such that $(a+bi)(x+yi)=(1+0i)$

$(a+bi)(x+yi)=(ax-by)+(ay+bx)i=1+0i$

$ax-by=1$ (1)

$bx+ay=0$ (2)

$a,b$ are given

$x,y$ unknowns

• $b \times (1)$ $abx-b^2y=b$

• $a \times (2)$ $abx+a^2y=0$

$\Rightarrow a^{2}y+b^{2}y=-b$

$y=\frac{-b}{a^{2}+b^{2}}$

$x=\frac{a}{a^{2}+b^{2}}$

• (Note: We can divide since we assumed that $(a,b) \neq 0$

$(a+bi)^{-1}=\frac{a}{a^{2}+b^{2}}+\frac{-b}{a^{2}+b^{2}}i=\frac{a-bi}{a^{2}+b^{2}}=\frac{\overline{a+bi}}{|a+bi|^{2}}=\frac{\mbox{conjugate}}{\mbox{norm squared }}$

Def: Let $\mathbb C$ be the set of all pairs of real numbers ${(a,b)}={a+bi}$

with $+: (a,b)+(c,d)=(a+c,b+d)$

$(a+bi)+(c+di)=(a+c)+(b+d)i$

$\times :(a+bi)(c+di)=$...you know what

• 0 = you know what

• 1 = you know what

Theorem:

1. $\mathbb C$ is a field

1. $(0+1i)^2=(0,1)^2=i^2=-1_{C}=(-1,0)$

1. $\mathbb R \rightarrow \mathbb C$ by $a \rightarrow a+0i$

Proof: $F_{1},F_{2},F_{3},...$

Example: $F_{5}$ (distributivity)

• Show that $z(u+v)=zu+zv$

Let $z=(a+bi)$

$u=(c+di)$

$v=(e+fi)$

When $a,b,c,d,e,f \in \mathbb R$

$(a+bi)[(c+di)+(e+fi)]=(a+bi)(c+di)+(a+bi)(e+fi)=(ac-bd)+\ldots$

• NEXT WEEK: Complex numbers have geometric meaning, geometric interpretation (waves)