sqrtPR

sqrtPR is a mathematical notation used to denote the square root of the product of two quantities, commonly written as sqrt(PR). It is defined for real numbers P and R in contexts where the product PR is nonnegative, or more generally for complex numbers with the appropriate branch choice of the square root.

When P and R are nonnegative real numbers, sqrt(PR) equals the product of their square roots: sqrt(PR) = sqrt(P) sqrt(R). This follows from the properties of the square root function and is often convenient in calculations. In this domain, sqrt(PR) also represents the geometric mean of P and R, up to the condition that P and R be positive; specifically, for positive P and R, sqrt(PR) is the geometric mean of the pair.

Key properties include monotonicity in each argument: increasing P or R (while keeping the other fixed) does not decrease sqrt(PR). For nonnegative P and R, the inequality sqrt(PR) ≤ (P+R)/2 holds, with equality only when P = R, illustrating the relationship to arithmetic and geometric means.

In practical use, sqrtPR appears in formulas across mathematics, physics, statistics, and engineering whenever a magnitude proportional to the geometric mean of two quantities is required. In programming or scientific computation, it is usually computed as sqrt(P*R); care is needed to avoid overflow in the product, and to handle cases where P or R may be negative or complex, which changes the interpretation of the square root.

See also: square root, geometric mean, product, AM-GM inequality.