taking square root algebraically

For getting the square root of the complex numberMathworldPlanetmathPlanetmatha+ib  (a,b) purely algebraically, one should solve the real partMathworldPlanetmath x and the imaginary part y of  a+ib  from the binomial equation

(x+iy)2=a+ib. (1)

This gives


Comparing (see equality (http://planetmath.org/EqualityOfComplexNumbers)) the real parts and the imaginary parts yields the pair of real equations


which may be written


Note that the x and y must be chosen such that their productPlanetmathPlanetmath (=b2) has the same sign as b.  Using the properties of quadratic equation, one infers that x2 and -y2 are the roots of the equation


The quadratic formula gives


and since -y2 is the smaller root,  x2=a+a2+b22,-y2=a-a2+b22.  So we obtain the result


(see the signum function).  Because both may have also the

a+ib=±(a2+b2+a2+(signb)ia2+b2-a2). (2)

The result shows that the real and imaginary parts of the square root of any complex number  a+ib  can be obtained from the real part a and imaginary part b of the number by using only algebraic operations, i.e. the rational operationsMathworldPlanetmath 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 i and check it!

Title taking square root algebraically
Canonical name TakingSquareRootAlgebraically
Date of creation 2015-06-14 16:31:35
Last modified on 2015-06-14 16:31:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Derivation
Classification msc 30-00
Classification msc 12D99
Synonym square root of complex number
Related topic SquareRootOfSquareRootBinomial
Related topic CasusIrreducibilis
Related topic TopicEntryOnComplexAnalysis
Related topic ValuesOfComplexCosine