sqrtu2

sqrtu2 refers to the mathematical expression for the square root of u squared, commonly written as sqrt(u^2). In real arithmetic, sqrt(u^2) equals the absolute value of u, denoted |u|. This means sqrt(u^2) = u when u is nonnegative, and sqrt(u^2) = -u when u is negative. The result is always nonnegative, reflecting the definition of the principal square root.

In the complex plane, the identity sqrt(z^2) = |z| does not hold in general because of the branch choice for the complex square root. The principal value sqrt(z) is defined with a branch cut, and sqrt(z^2) may equal z or -z depending on z and the chosen branch. A simple example: for z = -1, sqrt((-1)^2) = sqrt(1) = 1, not -1.

When simplifying expressions with sqrt(u^2) in real algebra, replace with |u| to avoid ambiguity. Squaring both sides of an equation and then taking square roots can introduce extraneous solutions; using the identity sqrt(u^2) = |u| helps prevent this.

Graphically, y = sqrt(u^2) is the absolute value function y = |u|, a V-shaped curve that is nonnegative and symmetric about the origin. It also appears in distance formulas such as sqrt(a^2 + b^2).

In summary, sqrtu2 encodes the principal square root of a squared real quantity, equal to the absolute value in real numbers, with additional caveats in complex analysis.