Square Roots - how to find them?

A square root of a is defined as a solution of the equation

x^2 - a = 0

over some field. Solving this equation, which operation we call "finding the square root", is an Algebraic problem.

Depending upon the field and the specific element a, therein, there might be no solution, one (that is, a unique) solution, a pair (that is, conjugate) solutions, or more. If a is an integer and a solution exists, then it is either an integer or an algebraic irrational number. In the field of real numbers, there are a pair of conjugate solutions if a > 0, one solution if a = 0, and none if a < 0. In the field of complex numbers, to which we will confine ourselves, henceforth, there always exist a pair of roots, whose sum is zero. They are complex-conjugate iff (= if and only if) the original number is real-complex and negative.

Finding a square-root is just a special case of solving a polynomial equation. Hence, we provide some background information on polynomials.

There are numerous methods for finding a square root. A transcendental method is considered as being "over-kill", thus less elegant. Some of the obvious methods for finding a square root are

binary search. It will find the exact solution, if such exists; otherwise it is an iterative procedure. This method requires ordering of the domain of the function; thus, since complex numbers are not ordered, this method does not work for a complex domain. The binary search starts well; but, compared to the Raphson and Newton methods, it converges slowly.
binomial theorem. While this is an infinite series, it is not iterative. This method is applicable to finding any real root of a number.
Horner's method. finds the roots of any polynomial equation. It will find the exact solution, if such exists; otherwise it is an iterative procedure. To find the two square roots of a, solve the algebraic polynomial equation x^2 - a = 0, employing the Horner's method. The Horner's method is the preferred method for hand solution.
Raphson method. This is an iterative procedure. This method is applicable to any function which has a few derivatives. The Raphson method does not start by itself; but converges faster than the binary search method. Ordinarily, one would employ the binary search method to start the Raphson method.
Newton's method. This is an iterative procedure. The Newton's method converges faster than either the binary search or the Raphson methods. While the Newton's method may start itself; it does not do so reliably. Ordinarily, one would employ the Raphson method to start the Newton's method. The Newton's method is the preferred method for use by computers, at least for real roots.
Taylor's series. While this is an infinite series, it is not iterative. This method is applicable to any function which has all derivatives.
logarithm and exponential. This clearly is a transedental method. Of itself, it requires only two steps: sqrt(x) = exp((1 / 2) ln(x)). But, we are left with the new problems of finding a logarithm and an exponential. This method is prefared for finding roots of complex numbers.

Both infinite series and iterative procedures can be made to yield a value arbitrarily close to the root, if enough steps are taken. Each of these methods is more general than just for finding a square root. However, while each of their scopes include finding of a square root, they are not identical. Also, they differ in their ease of use for hand calculation.

Of course, if you either look-up the square root in a table or employ a calculator or computer program to compute the square root, you just are hiding the method from your own view.

What about if the number whose square-root is desired is complex? With more or less effort, most of the foregoing methods may be adapted. However, the following theorem shows that the problem is reducible to that of three real square-roots.

Theorem: Complex square-root. The square-root of the complex number (a + i b) is sqrt(a + i b) = +- (sqrt(r + a) + i sqrt(r - a) sign(b)) sqrt(2) / 2, where r = sqrt(a^2 + b^2).

Proof: Hint: Square each side.