Give a corect answer for this question how do I

eozoischgc

eozoischgc

Answered question

2022-01-12

Give a corect answer for this question how do I get the square root of a complex number? If given a complex number (say 9+4i), how do I calculate its square root?

Answer & Explanation

usaho4w

usaho4w

Beginner2022-01-13Added 39 answers

The square root is not a well defined function on complex numbers. Because of the fundamental theorem of algebra, you will always have two different square roots for a given number. If you want to find out the possible values, the easiest way is probably to go with De Moivres
Timothy Wolff

Timothy Wolff

Beginner2022-01-14Added 26 answers

Here is a direct algebraic answer.
Suppose that z=c+di, and we want to find z=a+bi lying in the first two quadrants. So what are a and b?
Precisely we have
a=c+c2+d22
and
b=d|d|c+c2+d22
(The factor of d|d| is used so that b has the same sign as d) To find this, we can use brute force and the quadratic formula. Squaring, we would need to solve
a2b2+2abi=c+di.
This gives two equations and two unknowns (separate into real and imaginary parts), which can then be solved by substitutions and the quadratic formula.
nick1337

nick1337

Expert2022-01-14Added 777 answers

You can also do following (technique often advised at school) :
Let's write z2=9+4i with z=a+bi. The goal is to find z
Thus we have (a+bi)2=9+4i and if you expend we get a2+2abib2=9+4i
If you identify the real and imaginary parts, you obtain :
a2b2=9 (1)
and
2ab=4 (2)
Now, as z2=9+4i, the modulus of z2 and 9+4i are equal so we can write :
a2+b2=92+42
a2+b2=97 (3)
Now find a and b with the the equations (1) , (2) and (3) :
K(1)+(3)2a2=9+97
so a=12(9+97) or a=12(9+9)
With equation (2) and the previous result you can now find b :
2ab=4
b=2/a
so b=229+97orb=229+97
The answer is : z=12(9+97)+2i29+97orz=12(9+97)2i29+97

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