Expert: Alon Mandes - 8/16/2009

Question
QUESTION: The complex number z is given by z=(4-3i)/(1-2i). Find the two square roots of complex number 5-12i, giving your answers in the form x+iy,where x & y are real. Many Thanks

ANSWER: √(5-12i) will give 2 solutions. ("Dividing the unit circle into 2 pieces"). Lets 1st write
the number 5-12i in polar form :
ρ=√5²+12²=√169=13
Ø=Arctan[-12/5]=Arctan[-2.4]=-67.38°
After performing the square root √ we get a new complex number with :
r=√ρ=√13 & 2 angles :
θ1=½Ø=½*-67.38°=-33.69°
θ2=½(Ø+360°)=½(-67.38+360°)=146.3° .
Now we want to transform the polar form into the cartesian form as x+iy. Thus :
1st solution : x+iy=rcos(θ1)+irsin(θ1)=rcos(-33.69°)+isin(-33.69°)=√13*0.832+i=√13*-0.554
2nd solution : x+iy=rcos(θ2)+irsin(θ2)=rcos(146.3°)+isin(146.3°)=√13*-0.381+i√13*0.554

1st solution : 2.99-1.99i
2nd solution : -1.37+1.99i

Alon.


---------- FOLLOW-UP ----------

QUESTION: But the given answers to me are 3-2i & the other is -3+2i. What does arctan means? Please explain arctan. Thanks. Do you mind checking your solutions again please & can you type your solutions using the symbols that can be found in mobile, please? Because I can't view some of the mathematics symbols. Much appreciated. Thank you very much

Answer
Ok,
First of all, I had a slight mistake  in the last row :
2nd solution : rcos(146.3°)+isin(146.3°)=√13*-0.381+i√13*0.554 .. It should be :
2nd solution : rcos(146.3°)+isin(146.3°)=√13*-0.831+i√13*0.554 .
cos(146.3°)=-0.831 & not -0.381 !!!! & for that I apologize .
Thus, the solutions will be :
1st solution : 2.99-1.99i
2nd solution : -2.99+1.99i

Second, I got 2.99 & not 3 because I took three numbers after the decimal point. If you take more & more you will get 2.9999 & even 3 .

Alon.