taking square root algebraically
Comparing (see equality (http://planetmath.org/EqualityOfComplexNumbers)) the real parts and the imaginary parts yields the pair of real equations
which may be written
The quadratic formula gives
and since is the smaller root, . So we obtain the result
(see the signum function). Because both may have also the
The result shows that the real and imaginary parts of the square root of any complex number can be obtained from the real part and imaginary part of the number by using only algebraic operations, i.e. the rational operations and the . Apparently, the same is true for all roots of a complex number with index (http://planetmath.org/NthRoot) an integer power of 2.
In practise, when determining the square root of a non-real complex number, one need not to remember the (2), but it’s better to solve concretely the equation (1).
Exercise. Compute and check it!
|Title||taking square root algebraically|
|Date of creation||2015-06-14 16:31:35|
|Last modified on||2015-06-14 16:31:35|
|Last modified by||pahio (2872)|
|Synonym||square root of complex number|