SOLUTION: Find the modulus and argument of √(1+2i)/(1-2i)

Did you mean this:  or this: ?

I'll assume the second



















let









Setting real parts equal and imaginary parts equal.





Solve that system and get a = ±1, b = ±2

So there are two solutions to the system: ±1±2i

In either case the modulus is 

But the modulus must be multiplied by the factor 

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In the case of 1+2i, which is in the first quadrant with 





In the case of -1-2i, which is in the third quadrant with 





Edwin

So, we consider  .    (1)


First consider the ratio under the square root  .    (2)


The numerator has the modulus  |1+2i| =  = .

The denominator has the modulus  |1-2i| =  = .


Therefore, the ratio    has the modulus of   = 1.


Hence, the original complex number   has the modulus   = 1.


    +-----------------------------------------------+
    |    Thus we completed with the modulus.        |
    |    Now, let's determine the argument of (1).  |
    +-----------------------------------------------+



Consider the ratio under the square root   .    (3)


The numerator has the argument "a" in QI such that  tan(a) = 2/1 = 2.

The denominator has the argument "b" in QIV such that tan(b) = (-2)/1 = -2.

Therefore, b = -a  (which is obvious).

Hence, the ratio    has the argument  a - b = a - (-a) = 2a.


When we take square root of (3), the argument of the resulting complex number will be  = a.


Thus we get the ANSWER: 

    +----------------------------------------------------------------------------+
    |    the modulus of the sough complex number is 1;                           |
    |                                                                            |
    |    the argument of the sought complex number is "a" such that tan(a) = 2,  |
    |        i.e. a = arctan(2) = 1.10715 radians = 63.435 degrees (rounded).    |     
    +----------------------------------------------------------------------------+


For completeness, when we take square root of (3), we obtain, actually, two complex numbers.
First one has the argument "a", as we found it above; the second complex number has the argument a+.

So, the second value of the square root has the modulus of 1 and the argument arctan(2)+ = 243.435 degrees.