What is the easy method to find the square roots of natural numbers?
Answers
Step-by-step explanation:
Now you have two numbers that multiply to get your original number. Take the average of these two numbers. This becomes your second guess for the square root. So again, you can divide the original number by this new guess, and take the average of these two numbers to get a third guess, and so on.
Step-by-step explanation:
Methods of computing square roots are numerical analysis algorithms for finding the principal, or non-negative, square root (usually denoted √S, 2√S, or S1/2) of a real number. Arithmetically, it means given S, a procedure for finding a number which when multiplied by itself, yields S; algebraically, it means a procedure for finding the non-negative root of the equation x2 - S = 0; geometrically, it means given the area of a square, a procedure for constructing a side of the square.
Every real number has two square roots.[Note 1] The principal square root of most numbers is an irrational number with an infinite decimal expansion. As a result, the decimal expansion of any such square root can only be computed to some finite-precision approximation. However, even if we are taking the square root of a perfect square integer, so that the result does have an exact finite representation, the procedure used to compute it may only return a series of increasingly accurate approximations.
The continued fraction representation of a real number can be used instead of its decimal or binary expansion and this representation has the property that the square root of any rational number (which is not already a perfect square) has a periodic, repeating expansion, similar to how rational numbers have repeating expansions in the decimal notation system.
The most common analytical methods are iterative and consist of two steps: finding a suitable starting value, followed by iterative refinement until some termination criteria is met. The starting value can be any number, but fewer iterations will be required the closer it is to the final result. The most familiar such method, most suited for programmatic calculation, is Newton's method, which is based on a property of the derivative in the calculus. A few methods like paper-and-pencil synthetic division and series expansion, do not require a starting value. In some applications, an integer square root is required, which is the square root rounded or truncated to the nearest integer (a modified procedure may be employed in this case).
The method employed depends on what the result is to be used for (i.e. how accurate it has to be), how much effort one is willing to put into the procedure, and what tools are at hand. The methods may be roughly classified as those suitable for mental calculation, those usually requiring at least paper and pencil, and those which are implemented as programs to be executed on a digital electronic computer or other computing device. Algorithms may take into account convergence (how many iterations are required to achieve a specified precision), computational complexity of individual operations (i.e. division) or iterations, and error propagation (the accuracy of the final result).