Complex numbers

What do you want to know about them?

 

Well, the procedure is essentially identical to converting between rectangular and polar coordinates on the good ol' real plane, so if that's where you're having trouble, you can pick up one of your old textbooks and review.

 

As Hurkyl said, just think of the good ole plane here.

Examples:
Suppose that a complex number z is given by:
z=a+ib
where a,b are real numbers, and i the imaginary unit.
Then, multiply z with 1 in the following manner:
[tex]z=\frac{\sqrt{a^{2}+b^{2}}}{\sqrt{a^{2}+b^{2}}}(a+ib)={\sqrt{a^{2}+b^{2}}}(\frac{a}{\sqrt{a^{2}+b^{2}}}+i\frac{b}{\sqrt{a^{2}+b^{2}}})[/tex]
Find the angle [tex]\theta[/tex] that is the solution of the system of equations:
[tex]\frac{a} {\sqrt{a^{2}+b^{2}}}=\cos\theta,\frac{b}{\sqrt{a^{2}+b^{2}}}=\sin\theta[/tex]
Thus, defining [tex]|z|={\sqrt{a^{2}+b^{2}}}[/tex], we get:
[tex]z=|z|(\cos\theta+i\sin\theta)=|z|e^{i\theta}[/tex]
by definition of the complex exponential.

 

buy a ti 89 or voyage 200 and your problems are forever solved

 

Maybe these websites will be a little help for you:

http://mathworld.wolfram.com/ComplexNumber.html
http://www.clarku.edu/~djoyce/complex/polar.html

I can only give you one tip:
(i) Get familiar with graphical interpretation of the sine and cosine in a circle.
(ii) Really try to understand the formula by examining the drawing of a complex number (like in the link above).

[tex]z=|z|(\cos\theta+i\sin\theta)=|z|e^{i\theta}[/tex]

 

Euler Identity

Here is a cool trick for calculating pi derived from Euler Identity.

e^(i*(pi/2)) = Cos(90) + i*Sin(90)

ln(e^(i*(pi/2)) = ln(Cos(90) +i*Sin(90))

i*(pi/2)*lne = ln(Cos(90) +i*Sin(90))

pi = (1/i)*(2)*ln(Cos(90) +i*Sin(90))

pi = (i^4/i)*(2)*ln(Cos(90) +i*Sin(90))

pi = (-i)*(2)*ln(Cos(90) +i*Sin(90))

pi = (-2i)*ln(Cos(90) +i*Sin(90))

pi = ln((Cos(90) +i*Sin(90))^(-2i))

pi = ln(1/(Cos(90) +i*Sin(90))^(2i))

Just a cool trick!

Best Regards,

Edwin G. Schasteen

 

"buy a ti 89 or voyage 200 and your problems are forever solved"

if his problem is understanding how certain things work, then i think his problem would stay untouched if he bought one of these caluclators.

 

I have a TI-83 plus and a TI Voyage 200, and I carry them both with me everywhere I go! They are truely amazing computation devices for those of us that are numerically challenged or just plain lazy.

Best Regards,

Edwin