sqrt34

sqrt34 denotes the positive square root of 34. It is an irrational real number, since 34 is not a perfect square. Its decimal expansion begins 5.830951894845301... and continues without termination or repetition.

Algebraically, sqrt(34) is a root of the monic polynomial x^2 − 34 = 0, making it an algebraic integer of degree 2 over the rational numbers. In the real quadratic field Q(√34), its algebraic conjugate is −√34. The pair {√34, −√34} reflects the two embeddings of this field into the real and complex numbers.

Geometrically, √34 is the hypotenuse of a right triangle with legs of lengths 3 and 5, since 3^2 + 5^2 = 34. This simple example illustrates how irrational square roots arise naturally in distance calculations.

In arithmetic terms, √34 squared yields 34, and the integer solutions to a^2 + b^2 = 34 include (±3, ±5) and (±5, ±3). Like other square roots of non-square integers, √34 has a periodic continued fraction representation, a hallmark of quadratic irrationals, though its exact period is a property of its continued fraction expansion.

See also: irrational numbers, quadratic surds, square roots, continued fractions.