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 following properties of the square root functions are important:
- √x = x1/2, which can be used to find its derivative using the power rule.
- √(x · y) = √x · √y
- √(x / y) = √x / √y
- √(x2) = |x| for every real number x (see absolute value)
The square root function generally maps rational numbers to algebraic and constructible irrational numbers, except in cases where the numerator and denominator are both perfect squares. It also maps the area of a square to its side length.
Rapidly convergent methods for calculating rational approximations of irrational square roots include Newton's method and Pell's equation. The algorithm based on Newton's method for approximating √x proceeds as follows:
- start with an arbitrary positive start value r (the closer to the root the better)
- replace r by the average of r and x/r
- go to 2
This is a quadratically convergent algorithm, which means that the number of correct digits of r roughly doubles with each step.