sqrt7

sqrt(7) denotes the positive real number whose square equals 7. It is irrational, meaning it cannot be expressed as a ratio of integers. It is an algebraic number of degree 2, a root of the polynomial x^2 − 7 = 0, and it lies in the quadratic field Q(√7).

Because 7 is not a perfect square, sqrt(7) is a quadratic irrational. Its continued fraction expansion is periodic: sqrt(7) = [2; 1,1,1,4,1,1,1,4, ...], repeating the block 1,1,1,4 with period length 4. In decimal form, sqrt(7) ≈ 2.64575131..., and common approximations such as 2.6458 are used when higher precision is unnecessary.

An elementary property is its irrationality: if sqrt(7) = p/q in lowest terms, then p^2 = 7q^2 leads to 7 dividing p, which forces 7 dividing q as well, a contradiction. Thus sqrt(7) cannot be rational.

In number theory, sqrt(7) appears in Pell-type equations, notably x^2 − 7y^2 = 1. The fundamental solution is (x, y) = (8, 3), since 8^2 − 7·3^2 = 1. All positive solutions arise from powers of (8 + 3√7) in the ring Z[√7].

Because it is not constructible from rational numbers by a finite sequence of square roots, sqrt(7) serves as a standard example of a simple quadratic irrational with a periodic continued fraction and explicit connections to Pell equations.

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