sqrt6857

The square root of 6857, written as sqrt(6857) or √6857, is the positive real number x that satisfies x^2 = 6857. It is an irrational number because 6857 is not a perfect square.

6857 is an odd integer and, when verified for divisibility by primes up to its square root, is not divisible by any of them; in particular, it is considered a prime number. Consequently, sqrt(6857) is an algebraic number of degree 2 over the rational numbers, meaning it cannot be expressed as a ratio of integers.

Numerical value and accuracy: sqrt(6857) is approximately 82.8070045. A common rounded value is 82.8070 or 82.807 when a few decimal places are sufficient. The square of this number is exactly 6857 within the limits of the rounding used in computations.

Mathematical properties: As the square root of a non-square integer, sqrt(6857) has a non-terminating, non-repeating decimal expansion. The continued fraction expansion of sqrt(6857) is periodic, a property shared by the square roots of all non-square integers. This periodicity underlies methods for finding good rational approximations to sqrt(6857), such as convergents in its continued fraction expansion.

Context and usage: Values like sqrt(6857) appear in various areas of number theory and algorithmic computation, including problems involving Pell-type equations and the study of irrationality and approximations to quadratic surds.