sqrt21

sqrt(21) denotes the positive square root of 21. It is an irrational real number and an algebraic number of degree 2, since it satisfies the quadratic equation x^2 = 21. Its minimal polynomial over the rationals is x^2 − 21. The two algebraic conjugates are sqrt(21) and −sqrt(21). In decimal form, sqrt(21) ≈ 4.582575695. It can be approximated using methods such as Newton-Raphson; convergents of its continued fraction provide increasingly accurate rational approximations.

In the theory of numbers, sqrt(21) generates the real quadratic field Q(sqrt(21)). The ring of integers in this field is Z[(1+sqrt(21))/2], since 21 ≡ 1 (mod 4). The continued fraction expansion of sqrt(21) is periodic, a common feature of square roots of non-square integers. Solutions to Pell equations x^2 − 21 y^2 = 1 are related to powers of sqrt(21); the fundamental solution is (x, y) = (55, 12).