Square root

In mathematics, the square root of a non-negative real number x is that non-negative real number which, when multiplied by itself, gives x. The square root of x is denoted by √x. For example, √16 = 4 since 4 × 4 = 16, and √2 = 1.41421... . Square roots are important when solving quadratic equations. Trying to extend the square root function to the negative numbers leads to imaginary numbers and eventually to the field of complex numbers.

The square root symbol was first used during the 16th Century. It has been suggested that it originated as an altered form of lowercase r, representing the Latin "radix" (meaning "root").

Table of contents

1 Properties 2 Computing square roots 2.1 Calculators 2.2 Babylonian method 2.3 An exact "long-division like" algorithm 2.4 Pell's equation 2.5 Continued fraction methods 3 Square roots of complex numbers 4 Square roots of matrices and operators 5 Square roots of the first 20 positive integers

Properties

The following important properties of the square root functions are valid for all positive real numbers x and y:

for every real number x (see absolute value)

The square root function generally maps rational numbers to algebraic numbers; √x is rational if and only if x is a rational number which, after cancelling, is a fraction of two perfect squares. In particular, √2 is irrational.

The square root function also maps the area of a square to its side length.

Suppose that x and a are reals, and that x²=a, and we want to find x. A common mistake is to "take the square root" and deduce that x = √a. This is incorrect, because the square root of x² is not x, but the absolute value |x|, one of our above rules. Thus, all we can conclude is that |x| = √a, or equivalently x = ±√a.

In calculus, for instance when proving that the square root function is continuous or differentiable or when computing certain limitss, the following identity often comes handy:

It is valid for all non-negative numbers x and y which are not both zero.

The function f(x) = √x has the following graph:

The function is continuous for all non-negative x, and differentiable for all positive x (it is not differentiable for x=0 since the slope of the tangent there is &infin). Its derivative is given by

Its Taylor series about x = 1 can be found using the binomial theorem:

for |x| < 1.

Computing square roots

Calculators

Pocket calculatorss typically implement good routines to compute the exponential function and the natural logarithm, and then compute the square root of x using the identity

The same identity is exploited when computing square roots with logarithm tables or slide rules.

Babylonian method

A commonly used algorithm for approximating √x is known as the "Babylonian method" and is based on Newton's method. It proceeds as follows:

1. start with an arbitrary positive start value r (the closer to the root the better)
2. replace r by the average of r and x/r
3. go to 2

This is a quadratically convergent algorithm, which means that the number of correct digits of r roughly doubles with each step.

This algorithm works equally well in the p-adic numbers, but cannot be used to identify real square roots with p-adic square roots; it is easy, for example, to construct a sequence of rational numbers by this method which converges to +3 in the reals, but to -3 in the 2-adics.