sqrt2b2

sqrt2b2 denotes the square root of the expression 2 times b squared, that is, the principal square root of 2 b^2. In standard notation it corresponds to sqrt(2 b^2). For real numbers b, this quantity is always defined and nonnegative.

A common simplification uses the identity sqrt(b^2) = |b|. Therefore sqrt(2 b^2) = sqrt(2) |b|. If b is known to be nonnegative, the expression further simplifies to sqrt(2) b. If b is negative, the value becomes sqrt(2) times the absolute value of b.

Domain and range: the domain is all real numbers b, since b^2 is nonnegative for all b. The range is the nonnegative real numbers, with 0 achieved only at b = 0. The graph of y = sqrt(2 b^2) is a V-shaped function, equivalent to y = sqrt(2) |b|, with vertex at (0, 0) and slopes ±sqrt(2).

Relation to norms: sqrt(2 b^2) can be viewed as sqrt(2) times the absolute value of b. It is proportional to the one-dimensional Euclidean norm of the vector (b) scaled by sqrt(2). More generally, sqrt(c b^2) = sqrt(c) |b| for any positive c.

Notation caveat: this expression should not be confused with sqrt(2) b^2, which would equal b^2 sqrt(2) and is not equivalent. In programming, explicit abs(b) is often used to preserve the correct signless magnitude.