|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
taking square root algebraically
|
(Derivation)
|
|
|
For getting the square root of the complex number $a\!+\!ib$ ($a,\,b\in \mathbb{R}$ ) purely algebraically, one should solve the real part $x$ and the imaginary part $y$ of $\sqrt{a\!+\!ib}$ from the binomial equation
 |
(1) |
This gives $$a\!+\!ib = x^2\!+\!2ixy\!-\!y^2 =(x^2\!-\!y^2)\!+\!i\!\cdot\!2xy.$$ Comparing (see equality) the real parts and the imaginary parts yields the pair of real equations $$x^2\!-\!y^2 = a,\quad 2xy = b,$$ which may be written $$x^2\!+\!(-y^2) = a, \quad x^2\!\cdot\!(-y^2) = -\frac{b^2}{4}.$$ Note that the signs of $x$ 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, $x^2 = \frac{a\!+\!\sqrt{a^2\!+\!b^2}}{2},\quad -y^2 = \frac{a\!-\!\sqrt{a^2\!+\!b^2}}{2}$ . So we obtain the result $$x = \sqrt{\frac{\sqrt{a^2\!+\!b^2}+\!a}{2}}, \quad y = (\mbox{sign}\,b)\sqrt{\frac{\sqrt{a^2\!+\!b^2}-\!a}{2}}$$ (see the signum function). Because both may have also the opposite signs, we have the formula
 |
(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 radicals. Apparently, the same is true for all roots of a complex number with index an integer power of 2.
In practise, when determining the square root of a non-real complex number, one need not to remember the formula (2), but it's better to solve concretely the equation (1).
Exercise. Compute $\sqrt{i}$ and check it!
|
"taking square root algebraically" is owned by pahio.
|
|
(view preamble | get metadata)
Cross-references: power, integer, rational, operations, algebraic, number, signum function, quadratic formula, roots, properties of quadratic equation, product, equations, real, binomial equation, imaginary part, real part, complex number, square root
There are 4 references to this entry.
This is version 11 of taking square root algebraically, born on 2005-06-20, modified 2008-01-29.
Object id is 7175, canonical name is TakingSquareRootAlgebraically.
Accessed 4855 times total.
Classification:
| AMS MSC: | 12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous) | | | 30-00 (Functions of a complex variable :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
