realsquareroot

The realsquareroot, commonly written sqrt(x), is the nonnegative real number y that satisfies y^2 = x. It is the principal square root function for real inputs, returning a nonnegative result for nonnegative x and undefined for negative x within the real numbers.

For real numbers, the realsquareroot is defined only when x is greater than or equal to zero. Its range is the set of nonnegative real numbers [0, ∞). The realsquareroot is the inverse of the squaring function restricted to the nonnegative domain: if y = sqrt(x) with y ≥ 0, then y^2 = x and x = y^2. This makes sqrt(x^2) equal to |x|, illustrating that squaring loses sign information.

Key properties include:

- Nonnegativity: sqrt(x) ≥ 0 for all x ≥ 0.

- Monotonicity: sqrt(x) is increasing on [0, ∞).

- Identity: sqrt(a b) = sqrt(a) sqrt(b) for nonnegative a and b; sqrt(a + b) ≤ sqrt(a) + sqrt(b).

- Exponent form: sqrt(x) equals x^(1/2) in real arithmetic.

Computation and notation:

- The function is denoted sqrt(x) or x^(1/2). For most real numbers, sqrt(x) is irrational or rational depending on x, with simple values like sqrt(0)=0, sqrt(1)=1, sqrt(4)=2, and sqrt(2) ≈ 1.4142.

- In numerical work, Newton’s method or other iterative schemes are used to approximate sqrt(x) when a closed form is not available.

Historical and extended context:

While the realsquareroot is defined for nonnegative real numbers, extensions to complex numbers yield a complex square root, with two possible values and a branch cut to manage ambiguity.