sqrtEx2

sqrtEx2 is a mathematical operator defined as the principal square root of the square of a number. For real inputs, it is equivalent to the absolute value function: sqrtEx2(x) = sqrt(x^2) = |x|. As a result, the output is always nonnegative, even if the input is negative.

In terms of basic properties, sqrtEx2 is idempotent: applying it twice yields the same result as applying it once, since sqrtEx2(sqrtEx2(x)) = sqrt(|x|^2) = |x|. It also satisfies the inequality sqrtEx2(x) ≥ 0 for all real x and acts as a magnitude operator, returning the distance of x from zero on the real line.

For real numbers, sqrtEx2 coincides with the absolute value, linking it to standard algebraic operations and geometric interpretations of length. Its derivative is piecewise defined: it is 1 for x > 0, -1 for x < 0, and not differentiable at x = 0, reflecting the cusp of the absolute value function. In programming, sqrtEx2 can be implemented as a conditional: if x ≥ 0 then x else -x, which yields the magnitude of x.

Limitations arise when extending sqrtEx2 to complex inputs, where sqrt(z^2) is not generally equal to the modulus |z| due to multi-valued square roots. Therefore, sqrtEx2 is typically treated as a real-valued operator, or restricted to real inputs, to preserve the simple interpretation as a magnitude.

See also: absolute value, modulus, square root.