09-240/Classnotes for Thursday September 17

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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 $\mathbb {C}$

• $=(a+c)+(bi+di)$

• $=(a+c)+(b+d)i$

• $=e+fi$

• $(a+bi)(c+di)=(a+c)+(b+d)i$

• $=a(c+di)+bi(c+di)$

• $=ac+adi+bic+bidi$

• $=ac+bdi^{2}+adi+bci$

• $=(ac-bd)+(ad+bc)i$

• $=e+fi$

• $0_{C}=0+0i$

• $1_{C}=1+0i$

• $(a+bi)+(c+di)=0+0i$

(rest of notes will be added in 1/2-hour)