Melody

Thanks Chris, that is really sweet of you :))    Thanks Dragonlance   :)

Hi Princess Rosala, it is nice to see you again:)

I found your likeness on the net, do you like it ?

This is the best Christmas skit ever !

I am going for Sir Isaac Newton because he is credited with inventing Calculus.

And I love Calculus!         

Hi Chris and Complex.

I did not check the first formula I just assumed it was correct.

Since the first one was a linear, and Chris's was linear, and if they were both correct they would HAVE to be equivalent.

So I assumed Chris's was the same as the one given. (I did not actually compare them)

Three points defines a line so all linear representions have to be equivalent. ://

m=2-sqrt(3)       

\(m.(4-m).(2-m)/sqrt(3)\\~\\ =\frac{m.(4-m).(2-m)}{\sqrt3}\\ \mbox{sub in }m=2-\sqrt3\\ =\frac{(2-\sqrt3)(4-(2-\sqrt3)).(2-(2-\sqrt3))}{\sqrt3}\\ =\frac{(2-\sqrt3)(2+\sqrt3).\sqrt3}{\sqrt3}\\ =(2-\sqrt3)(2+\sqrt3)\\ =4-3\\ =1\)

 (5 - 3i)^1/3.

\(r = mod z = |z| = \sqrt{5^2+(-3)^2}=\sqrt{34}\)

\(5-3i=\sqrt{34}(\frac{5}{\sqrt{34}}-\frac{3i}{\sqrt{34}})\\ 4th\;quad\\ argument \;of\; z = arg(z) =\theta=2\pi-acos(\frac{5}{\sqrt{34}})\\ \theta\approx 5.7427658\;radians\\ 5-3i=\sqrt{34}\;e^{5.7427658i} \)
 

\(|z^{1/3}| = \sqrt{34}^{1/3}=\sqrt[6]{34}\)

\(1)  arg(z^{1/3})=5.7427658/3 = 1.914255\\ 2)  arg(z^{1/3}) = 1.914255 +\frac{2 \pi}{3} = 4.008650\\ 3)  arg(z^{1/3}) = 1.914255 +2*\frac{2 \pi}{3}= 6.103045\\\)

So the 3 cubed roots of are 5-3i  are

\(\sqrt[6]{34}*e^{1.914255i}\quad and \quad \sqrt[6]{34}*e^{4.008650i}\quad and \quad \sqrt[6]{34}*e^{6.103045i}\)

cos(1.914255) = -0.336745771286

sin(1.914255) = 0.94159560615

\(1st\;root=\sqrt[6]{34}\;(-0.336746\;+\;0.941596\;i)\)

etc

check the first root

(34^(1/6)*e^(1.914255*i))^3 = 4.9999975792700509-3.0000040345460377i      Near enough    

Well to the nearest 10 

28 becomes 30

Is that what you mean?

What do you mean you found it? 

Do you mean you found it by yourself ?

Fri 25/12/15

Merry Christmas

https://web2.0calc.com/questions/merry-christmas-2015

1)  A Christmas Story from Dragonlance's Journal.  Very sweet and very funny, thanks Dragonlance.

     http://web2.0calc.com/questions/a-christmas-story-from-dragonlance-s-journal/new#edit

2)  Ouija Board Story by DragonLance:  Put this on your 'must read' list. It is hilarious.

     http://web2.0calc.com/questions/dragonlance-s-journal-stories#r1

3)  Mathematica Talent Contest       CPhill wins here !

     http://web2.0calc.com/questions/mathematica-talent-contest

4)  An interesting substitution queston.     Melody

     http://web2.0calc.com/questions/m-2-sqrt-3

5)  Easy complex number equation   Thanks Guest

     http://web2.0calc.com/questions/find-x-in-sqrt-1-x-2-i

6)  One more complex root , just for good measure

     http://web2.0calc.com/questions/complex-operation#r4

Hi Solveit :)

what does that ~ mean

I wasn't sure so I looked it up :)

It is used in geometry and it means "is similar to"

Two shapes are similar if they have the same angles and the sides are in the same proportions.

That is, one is an enlargement of the other one.  (It doesn't matter if the second one is turned or reflected)