sqrt105

The symbol √105 denotes the positive square root of the integer one hundred and five. Mathematically it is defined as the unique non‑negative real number x such that x² = 105. Because 105 is not a perfect square, its square root is an irrational number, meaning it cannot be expressed as a ratio of two integers. Consequently its decimal representation continues infinitely without repeating. The value of √105 is approximately 10.247223643. A more precise decimal expansion begins 10.247223643398538972505...

In terms of algebraic properties, √105 satisfies the minimal polynomial x² – 105 = 0 over the rational numbers. It thus belongs to the quadratic field ℚ(√105). This field is a real quadratic extension of ℚ and has ring of integers ℤ[√105] because the discriminant of the polynomial is 105, which is congruent to 1 mod 4. The norm of √105 in this field is –105, where the norm of a + b√105 is a² – 105b².

The square root of 105 arises in various contexts. In geometry, the diagonal of a cube of side length √105 would have length 105. It also appears in trigonometric identities and in certain algebraic expressions involving radicals, such as √(25+80) or √(36+69). In physics, √105 can appear in energy equations when combining constants that multiply to 105. The number does not have special significance in popular culture, but like all irrational square roots it serves as an example of the richness of the real number system.

The approximate value of √105 has been tabulated extensively in scientific handbooks and can be found by using the Newton–Raphson iteration method: starting from an estimate x₀, set x_{n+1} = (x_n + 105 / x_n) / 2, which converges rapidly to the true value. This process reflects the fundamental calculus idea that the function f(x) = x² – 105 has a simple root at √105.