# 09-240/Classnotes for Thursday September 17

## 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

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

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## Class notes for today

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

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

Implications:

${\displaystyle b\in \mathbb {R} \Rightarrow bi\in \mathbb {C} }$

${\displaystyle a\in \mathbb {R} \Rightarrow a+bi\in \mathbb {C} }$

${\displaystyle c,d\in \mathbb {R} \Rightarrow c+di\in \mathbb {C} }$

${\displaystyle \Rightarrow (a+bi)+(c+di)}$ must be in ${\displaystyle \mathbb {C} }$

${\displaystyle =(a+c)+(bi+di)}$
${\displaystyle =(a+c)+(b+d)i}$
${\displaystyle =e+fi}$

${\displaystyle (a+bi)(c+di)=(a+c)+(b+d)i}$

${\displaystyle =a(c+di)+bi(c+di)}$
${\displaystyle =ac+adi+bic+bidi}$
${\displaystyle =ac+bdi^{2}+adi+bci}$
${\displaystyle =(ac-bd)+(ad+bc)i}$
${\displaystyle =e+fi}$
${\displaystyle 0_{C}=0+0i}$
${\displaystyle 1_{C}=1+0i}$
${\displaystyle (a+bi)+(c+di)=0+0i}$
${\displaystyle -(a+bi)=(-a)+(-b)i}$
${\displaystyle a+bi\neq 0\Rightarrow (a,b)\neq 0}$

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

${\displaystyle (a+bi)(x+yi)=(ax-by)+(ay+bx)i=1+0i}$
${\displaystyle ax-by=1}$ (1)
${\displaystyle bx+ay=0}$ (2)
${\displaystyle a,b}$ are given
${\displaystyle x,y}$ unknowns

${\displaystyle b\times (1)}$ ${\displaystyle abx-b^{2}y=b}$

${\displaystyle a\times (2)}$ ${\displaystyle abx+a^{2}y=0}$

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

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

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

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

with ${\displaystyle +:(a,b)+(c,d)=(a+c,b+d)}$

${\displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i}$

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

• 0 = you know what

• 1 = you know what

Theorem:

1. ${\displaystyle \mathbb {C} }$ is a field
1. ${\displaystyle (0+1i)^{2}=(0,1)^{2}=i^{2}=-1_{C}=(-1,0)}$
1. ${\displaystyle \mathbb {R} \rightarrow \mathbb {C} }$ by ${\displaystyle a\rightarrow a+0i}$

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

Example: ${\displaystyle F_{5}}$ (distributivity)

• Show that ${\displaystyle z(u+v)=zu+zv}$

Let ${\displaystyle z=(a+bi)}$

${\displaystyle u=(c+di)}$
${\displaystyle v=(e+fi)}$

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

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

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