taking square root algebraically

For getting the square root of the complex number  $a+ib$ ($a,b\in ℝ$) purely algebraically, one should solve the real part $x$ and the imaginary part $y$ of  $\sqrt{a+ib}$  from the binomial equation

${(x+iy)}^2=a+ib.$(1)

This gives

$$a+ib=x^2+2ixy-y^2=(x^2-y^2)+i\cdot 2xy.$$

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

$$x^2-y^2=a,2xy=b,$$

which may be written

$$x^2+(-y^2)=a,x^2\cdot (-y^2)=-\frac{b^2}{4}.$$

Note that the $x$ and $y$ must be chosen such that their product ($=\frac{b}{2}$) has the same sign as $b$.  Using the properties of quadratic equation, one infers that $x^2$ and $-y^2$ are the roots of the equation

$$t^2-at-\frac{b^2}{4}=0.$$

The quadratic formula gives

$$t=\frac{a\pm \sqrt{a^2+b^2}}{2},$$

and since $-y^2$ is the smaller root,  $x^2=\frac{a+\sqrt{a^2+b^2}}{2},-y^2=\frac{a-\sqrt{a^2+b^2}}{2}$.  So we obtain the result

$$x=\sqrt{\frac{\sqrt{a^2+b^2}+a}{2}},y=(\mathrm{sign}b)\sqrt{\frac{\sqrt{a^2+b^2}-a}{2}}$$

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

$\sqrt{a+ib}=\pm \left(\sqrt{{\displaystyle \frac{\sqrt{a^2+b^2}+a}{2}}}+(\mathrm{sign}b)i\sqrt{{\displaystyle \frac{\sqrt{a^2+b^2}-a}{2}}}\right).$(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 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 $\sqrt{i}$ 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