sqrt2N

sqrt2n denotes the nonnegative square root of the product 2n, typically with n understood as a nonnegative real number or a natural number. In ordinary notation it is written as sqrt(2n). Because square roots distribute over multiplication, sqrt(2n) can be written as sqrt(2) sqrt(n). The expression is real for n ≥ 0.

Basic properties include monotonicity and squaring behavior: sqrt(2n) increases as n grows, and its square is 2n. The value is an integer exactly when 2n is a perfect square; since a perfect square that is even must have an even root, this occurs precisely when n = 2k^2 for some integer k, in which case sqrt(2n) = 2k. In all other cases, sqrt(2n) is irrational.

Asymptotically, sqrt(2n) grows proportionally to sqrt(n); more precisely, sqrt(2n) ~ sqrt(2) sqrt(n) for large n. This makes the expression common in contexts where a quantity scales with the square root of a parameter, such as geometric or probabilistic estimates, lattice point problems, and certain algorithmic analyses.

Examples include n = 1, yielding sqrt(2) ≈ 1.4142; n = 2, yielding sqrt(4) = 2; and n = 9, yielding sqrt(18) ≈ 4.2426. When n takes nonnegative real values, sqrt(2n) remains real; for n < 0 the expression is not real without extending to complex numbers.